A Framework for Constructive Category Theory in Computer Algebra [PDF]

R is a ring with identity, not necessarily commutative.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 22 / 65 1 ∈, =, 0, 1, +, −, ·, 2 solving one-sided linear systems of matrices with entries in R.

R is a ring with identity, not necessarily commutative. R is supposed to be computable, that is, we have algorithms for

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 22 / 65 2 solving one-sided linear systems of matrices with entries in R.

R is a ring with identity, not necessarily commutative. R is supposed to be computable, that is, we have algorithms for 1 ∈, =, 0, 1, +, −, ·,

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 22 / 65 RowsR is the category of matrices over R.

R is a ring with identity, not necessarily commutative. R is supposed to be computable, that is, we have algorithms for 1 ∈, =, 0, 1, +, −, ·, 2 solving one-sided linear systems of matrices with entries in R.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 22 / 65 Conventions

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 22 / 65 Motivation

1 Motivation

2 Homomorphism structures

3 Applications

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 23 / 65 That is, for given α and β ﬁnd ξ in the following diagram: P ξ α M N β That is, solve the one-sided linear system

ξ · β = α

for ξ.

Motivation Main aim

Compute lifts in fpresR

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 24 / 65 That is, solve the one-sided linear system

ξ · β = α

for ξ.

Motivation Main aim

Compute lifts in fpresR That is, for given α and β ﬁnd ξ in the following diagram: P ξ α M N β

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 24 / 65 Motivation Main aim

Compute lifts in fpresR That is, for given α and β ﬁnd ξ in the following diagram: P ξ α M N β That is, solve the one-sided linear system

ξ · β = α

for ξ.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 24 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that

means solving a two-sided linear system of matrices with entries in R. Why?

Solving a one-sided linear system in fpresR

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that

Why?

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

M N R1×m R1×n β hMi B hNi

R1×p P hPi ξ X α −→ A

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

R1×p hPi X −→ A

R1×m R1×n hMi B hNi

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

P ξ α M N β

R1×p hPi X A

R1×m R1×n hMi B hNi

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

P ξ α −→ M N β

1 ξ is well-deﬁned:

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00 X occurs both on the left and on the right.

1 ξ is well-deﬁned:

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 2 ξ · β = α: X · B = A +X 00 · N for some X 00 X occurs both on the left and on the right.

P · X = X 0 · M for some X 0

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 X · B = A +X 00 · N for some X 00 X occurs both on the left and on the right.

2 ξ · β = α:

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 X occurs both on the left and on the right.

X · B = A +X 00 · N for some X 00

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 X occurs both on the left and on the right.

+X 00 · N for some X 00

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 X occurs both on the left and on the right.

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 Motivation Problem

Solving a one-sided linear system in fpresR means solving a two-sided linear system of matrices with entries in R. Why?

Consider the data structure of fpresR: R1×p P hPi ξ X α −→ A

M N R1×m R1×n β hMi B hNi We have to ﬁnd X such that 1 ξ is well-deﬁned: P · X = X 0 · M for some X 0 2 ξ · β = α: X · B = A +X 00 · N for some X 00 X occurs both on the left and on the right.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 25 / 65 which concatenates the rows of a matrix M to get a long row vector. Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

× × Deﬁne vec: Qn m → Q1 nm

Aim Find a category theoretical abstraction of this trick.

For example, consider R = Q.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

which concatenates the rows of a matrix M to get a long row vector.

Aim Find a category theoretical abstraction of this trick.

For example, consider R = Q. × × Deﬁne vec: Qn m → Q1 nm

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

Aim Find a category theoretical abstraction of this trick.

For example, consider R = Q. × × Deﬁne vec: Qn m → Q1 nm which concatenates the rows of a matrix M to get a long row vector.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

Aim Find a category theoretical abstraction of this trick.

For example, consider R = Q. × × Deﬁne vec: Qn m → Q1 nm which concatenates the rows of a matrix M to get a long row vector. Then vec(M · X · N) = vec(X) · (MT ⊗ N)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Aim Find a category theoretical abstraction of this trick.

For example, consider R = Q. × × Deﬁne vec: Qn m → Q1 nm which concatenates the rows of a matrix M to get a long row vector. Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Aim Find a category theoretical abstraction of this trick.

A morphism in Category of left presentations of Z gap> IsCongruentForMorphisms( PreCompose( C^1, h0 ), phi[1] ); true gap> IsCongruentForMorphisms( PreCompose( h0, D^1 ), phi[0] ); true

Let S be a commutative ring. × × Deﬁne vec: Qn m → Q1 nm which concatenates the rows of a matrix M to get a long row vector. Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Aim Find a category theoretical abstraction of this trick.

Motivation Solution for commutative rings

Let S be a commutative ring. Deﬁne vec: Sn×m → S1×nm which concatenates the rows of a matrix M to get a long row vector. Then vec(M · X · N) = vec(X) · (MT ⊗ N) This explicitly depends on the commutativity of the ring (obvious for 1 × 1 matrices).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Motivation Solution for commutative rings

Aim Find a category theoretical abstraction of this trick.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 26 / 65 Homomorphism structures

1 Motivation

2 Homomorphism structures

3 Applications

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 27 / 65 A distinguished object 1 ∈ D A functor H : Cop × C → D ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

A D-homomorphism structure for C consists of the following data:

Moreover, if we are in the context of Ab-categories, we require H to be bilinear.

Homomorphism structures Homomorphism structures

Deﬁnition Let C and D be categories.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 A distinguished object 1 ∈ D A functor H : Cop × C → D ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C Moreover, if we are in the context of Ab-categories, we require H to be bilinear.

Homomorphism structures Homomorphism structures

Deﬁnition Let C and D be categories. A D-homomorphism structure for C consists of the following data:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 A functor H : Cop × C → D ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C Moreover, if we are in the context of Ab-categories, we require H to be bilinear.

Homomorphism structures Homomorphism structures

Deﬁnition Let C and D be categories. A D-homomorphism structure for C consists of the following data: A distinguished object 1 ∈ D

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C Moreover, if we are in the context of Ab-categories, we require H to be bilinear.

Homomorphism structures Homomorphism structures

Deﬁnition Let C and D be categories. A D-homomorphism structure for C consists of the following data: A distinguished object 1 ∈ D A functor H : Cop × C → D

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 Moreover, if we are in the context of Ab-categories, we require H to be bilinear.

Homomorphism structures Homomorphism structures

Deﬁnition Let C and D be categories. A D-homomorphism structure for C consists of the following data: A distinguished object 1 ∈ D A functor H : Cop × C → D ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 Homomorphism structures Homomorphism structures

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 28 / 65 Third property ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

This implies that the following diagram is commutative for all possible choices of α, β and ξ:

ξ ν ν(ξ)

HomC(α,β) HomD(id1,H(α,β)) α · ξ · β ν ν(α · ξ · β) = ν(ξ) · H(α, β)

Homomorphism structures Unwrapping the deﬁnition

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 29 / 65 This implies that the following diagram is commutative for all possible choices of α, β and ξ:

ξ ν ν(ξ)

HomC(α,β) HomD(id1,H(α,β)) α · ξ · β ν ν(α · ξ · β) = ν(ξ) · H(α, β)

Homomorphism structures Unwrapping the deﬁnition

Third property ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 29 / 65 ξ ν ν(ξ)

HomC(α,β) HomD(id1,H(α,β)) α · ξ · β ν ν(α · ξ · β) = ν(ξ) · H(α, β)

Homomorphism structures Unwrapping the deﬁnition

Third property ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

This implies that the following diagram is commutative for all possible choices of α, β and ξ:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 29 / 65 Homomorphism structures Unwrapping the deﬁnition

Third property ∼ An isomorphism ν : HomC(A, B) = HomD(1, H(A, B)) natural in A, B ∈ C

This implies that the following diagram is commutative for all possible choices of α, β and ξ:

ξ ν ν(ξ)

HomC(α,β) HomD(id1,H(α,β)) α · ξ · β ν ν(α · ξ · β) = ν(ξ) · H(α, β)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 29 / 65 C = D = RowsS H on morphisms: H(M, N) = MT ⊗ N H on objects: H(S1×n, S1×m) = S1×n·m ν(M) = vec(M) 1 = S1×1

Homomorphism structures Example

Let S be a commutative ring.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 H on morphisms: H(M, N) = MT ⊗ N H on objects: H(S1×n, S1×m) = S1×n·m ν(M) = vec(M) 1 = S1×1

Homomorphism structures Example

Let S be a commutative ring.

C = D = RowsS

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 H on objects: H(S1×n, S1×m) = S1×n·m ν(M) = vec(M) 1 = S1×1

Homomorphism structures Example

Let S be a commutative ring.

C = D = RowsS H on morphisms: H(M, N) = MT ⊗ N

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 ν(M) = vec(M) 1 = S1×1

Homomorphism structures Example

Let S be a commutative ring.

C = D = RowsS H on morphisms: H(M, N) = MT ⊗ N H on objects: H(S1×n, S1×m) = S1×n·m

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 1 = S1×1

Homomorphism structures Example

Let S be a commutative ring.

C = D = RowsS H on morphisms: H(M, N) = MT ⊗ N H on objects: H(S1×n, S1×m) = S1×n·m ν(M) = vec(M)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 Homomorphism structures Example

Let S be a commutative ring.

C = D = RowsS H on morphisms: H(M, N) = MT ⊗ N H on objects: H(S1×n, S1×m) = S1×n·m ν(M) = vec(M) 1 = S1×1

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 30 / 65 (c) 7→ (c) Naturality of ν: (a · c · b) = (c) · (a · b)

(S1×1, S1×1) 7→ S1×1 ((a), (b)) 7→ (a · b) Note: in this sense, H can be interpreted as the (enriched) Hom-functor on RowsS. 1×1 1×1 Similarly, ν is determined by its values on HomRowsS (S , S ):

Homomorphism structures Making use of the bilinearity

Since H is bilinear, it is already determined by its values on S1×1 1×1 1×1 and HomRowsS (S , S ):

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 (c) 7→ (c) Naturality of ν: (a · c · b) = (c) · (a · b)

((a), (b)) 7→ (a · b) Note: in this sense, H can be interpreted as the (enriched) Hom-functor on RowsS. 1×1 1×1 Similarly, ν is determined by its values on HomRowsS (S , S ):

Homomorphism structures Making use of the bilinearity

Since H is bilinear, it is already determined by its values on S1×1 1×1 1×1 and HomRowsS (S , S ): (S1×1, S1×1) 7→ S1×1

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 (c) 7→ (c) Naturality of ν: (a · c · b) = (c) · (a · b)

Note: in this sense, H can be interpreted as the (enriched) Hom-functor on RowsS. 1×1 1×1 Similarly, ν is determined by its values on HomRowsS (S , S ):

Homomorphism structures Making use of the bilinearity

Since H is bilinear, it is already determined by its values on S1×1 1×1 1×1 and HomRowsS (S , S ): (S1×1, S1×1) 7→ S1×1 ((a), (b)) 7→ (a · b)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 (c) 7→ (c) Naturality of ν: (a · c · b) = (c) · (a · b)

1×1 1×1 Similarly, ν is determined by its values on HomRowsS (S , S ):

Homomorphism structures Making use of the bilinearity

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 Naturality of ν: (a · c · b) = (c) · (a · b)

Homomorphism structures Making use of the bilinearity

Since H is bilinear, it is already determined by its values on S1×1 1×1 1×1 and HomRowsS (S , S ): (S1×1, S1×1) 7→ S1×1 ((a), (b)) 7→ (a · b) Note: in this sense, H can be interpreted as the (enriched) Hom-functor on RowsS. 1×1 1×1 Similarly, ν is determined by its values on HomRowsS (S , S ):

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 Naturality of ν: (a · c · b) = (c) · (a · b)

Homomorphism structures Making use of the bilinearity

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 Homomorphism structures Making use of the bilinearity

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 31 / 65 Solution: assume that we can ﬁnd a subring S of the center of R, such that R is ﬁnitely presented as an S-module. ∼ S1×m That is, SR = hMi . Then multiplication from left and right with elements of R is S-linear.

Homomorphism structures Non-commutative example

Problem: for a non-commutative ring and H and ν as above, we lose the naturality of ν.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 32 / 65 ∼ S1×m That is, SR = hMi . Then multiplication from left and right with elements of R is S-linear.

Homomorphism structures Non-commutative example

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 32 / 65 Then multiplication from left and right with elements of R is S-linear.

Homomorphism structures Non-commutative example

Problem: for a non-commutative ring and H and ν as above, we lose the naturality of ν. Solution: assume that we can ﬁnd a subring S of the center of R, such that R is ﬁnitely presented as an S-module. ∼ S1×m That is, SR = hMi .

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 32 / 65 Homomorphism structures Non-commutative example

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 32 / 65 1×1 1×1 S1×m (R , R ) 7→ hMi ((a), (b)) 7→ a · − · b expressed as a matrix in Sm×m

op H : RowsR × RowsR → fpresS:

S1×m ν is essentially the isomorphism ϕ: SR → hMi Naturality of ν: ϕ(a · c · b) = ϕ(c) · (a · − · b)

Homomorphism structures Non-commutative example (continued)

Using this, we can ﬁnd an fpresS-homomorphism structure (H, 1, ν) for RowsR:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 1×1 1×1 S1×m (R , R ) 7→ hMi ((a), (b)) 7→ a · − · b expressed as a matrix in Sm×m S1×m ν is essentially the isomorphism ϕ: SR → hMi Naturality of ν: ϕ(a · c · b) = ϕ(c) · (a · − · b)

Homomorphism structures Non-commutative example (continued)

Using this, we can ﬁnd an fpresS-homomorphism structure (H, 1, ν) for RowsR: op H : RowsR × RowsR → fpresS:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 ((a), (b)) 7→ a · − · b expressed as a matrix in Sm×m S1×m ν is essentially the isomorphism ϕ: SR → hMi Naturality of ν: ϕ(a · c · b) = ϕ(c) · (a · − · b)

Homomorphism structures Non-commutative example (continued)

Using this, we can ﬁnd an fpresS-homomorphism structure (H, 1, ν) for RowsR: op H : RowsR × RowsR → fpresS: 1×1 1×1 S1×m (R , R ) 7→ hMi

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 S1×m ν is essentially the isomorphism ϕ: SR → hMi Naturality of ν: ϕ(a · c · b) = ϕ(c) · (a · − · b)

Homomorphism structures Non-commutative example (continued)

Using this, we can ﬁnd an fpresS-homomorphism structure (H, 1, ν) for RowsR: op H : RowsR × RowsR → fpresS: 1×1 1×1 S1×m (R , R ) 7→ hMi ((a), (b)) 7→ a · − · b expressed as a matrix in Sm×m

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 Naturality of ν: ϕ(a · c · b) = ϕ(c) · (a · − · b)

Homomorphism structures Non-commutative example (continued)

Using this, we can ﬁnd an fpresS-homomorphism structure (H, 1, ν) for RowsR: op H : RowsR × RowsR → fpresS: 1×1 1×1 S1×m (R , R ) 7→ hMi ((a), (b)) 7→ a · − · b expressed as a matrix in Sm×m S1×m ν is essentially the isomorphism ϕ: SR → hMi

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 Homomorphism structures Non-commutative example (continued)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 33 / 65 In particular this works for the exterior algebra E of Qn over its center or over Q.

Homomorphism structures Remarks

As above, this data uniquely deﬁnes the homomorphism structure for all of RowsR.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 34 / 65 Homomorphism structures Remarks

As above, this data uniquely deﬁnes the homomorphism structure for all of RowsR. In particular this works for the exterior algebra E of Qn over its center or over Q.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 34 / 65 Applications

1 Motivation

2 Homomorphism structures

3 Applications

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 35 / 65 A linear system in C is a collection of morphisms αij , βij and γi of the following form:

α11 · X1 · β11 + ... +α1n · Xn · β1n = γ1 . .

αm1 · X1 · βm1 + ... +αmn · Xn · βmn = γm

where the Xj are unknown morphisms.

Applications Two-sided linear systems (1)

Let C and D be additive categories.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 36 / 65 Applications Two-sided linear systems (1)

Let C and D be additive categories.

A linear system in C is a collection of morphisms αij , βij and γi of the following form:

α11 · X1 · β11 + ... +α1n · Xn · β1n = γ1 . .

αm1 · X1 · βm1 + ... +αmn · Xn · βmn = γm

where the Xj are unknown morphisms.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 36 / 65 1 (ν(Xj ))j (ν(γi ))i L L i H(Ai , Di ) j H(Bj , Cj ) (H(αij ,βij ))ji

Using that we can transfer a two-sided linear system in C to a one-sided linear system in D by using ν(α · ξ · β) = ν(ξ) · H(α, β):

Applications Two-sided linear systems (2)

Assume we have a D-homomorphism structure for C.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 37 / 65 1 (ν(Xj ))j (ν(γi ))i L L i H(Ai , Di ) j H(Bj , Cj ) (H(αij ,βij ))ji

Applications Two-sided linear systems (2)

Assume we have a D-homomorphism structure for C. Using that we can transfer a two-sided linear system in C to a one-sided linear system in D by using ν(α · ξ · β) = ν(ξ) · H(α, β):

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 37 / 65 Applications Two-sided linear systems (2)

Assume we have a D-homomorphism structure for C. Using that we can transfer a two-sided linear system in C to a one-sided linear system in D by using ν(α · ξ · β) = ν(ξ) · H(α, β): 1 (ν(Xj ))j (ν(γi ))i L L i H(Ai , Di ) j H(Bj , Cj ) (H(αij ,βij ))ji

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 37 / 65 RowsS one-sided

RowsR fpresS two-sided one-sided

RowsS two-sided

Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 RowsS one-sided

fpresS one-sided

RowsS two-sided

Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

RowsR two-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 RowsS one-sided

RowsS two-sided

Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

RowsR fpresS two-sided one-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 RowsS one-sided

Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

RowsR fpresS two-sided one-sided

RowsS two-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

RowsR fpresS two-sided one-sided

RowsS RowsS two-sided one-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 RowsS one-sided

Applications

Lifts in fpresR

fpresR lift, i.e. one-sided

RowsR fpresS two-sided one-sided

RowsS one-sided

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 38 / 65 HomomorphismStructureOnMorphisms DistinguishedObjectOfHomomorphismStructure InterpretMorphismAsMorphismFromDinstinguished- ObjectToHomomorphismStructure InterpretMorphismFromDinstinguished- ObjectToHomomorphismStructureAsMorphism ⇒ SolveLinearSystemInAbCategory

Applications CAP basic operations

HomomorphismStructureOnObjects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 DistinguishedObjectOfHomomorphismStructure InterpretMorphismAsMorphismFromDinstinguished- ObjectToHomomorphismStructure InterpretMorphismFromDinstinguished- ObjectToHomomorphismStructureAsMorphism ⇒ SolveLinearSystemInAbCategory

Applications CAP basic operations

HomomorphismStructureOnObjects HomomorphismStructureOnMorphisms

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 InterpretMorphismAsMorphismFromDinstinguished- ObjectToHomomorphismStructure InterpretMorphismFromDinstinguished- ObjectToHomomorphismStructureAsMorphism ⇒ SolveLinearSystemInAbCategory

Applications CAP basic operations

HomomorphismStructureOnObjects HomomorphismStructureOnMorphisms DistinguishedObjectOfHomomorphismStructure

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 InterpretMorphismFromDinstinguished- ObjectToHomomorphismStructureAsMorphism ⇒ SolveLinearSystemInAbCategory

Applications CAP basic operations

HomomorphismStructureOnObjects HomomorphismStructureOnMorphisms DistinguishedObjectOfHomomorphismStructure InterpretMorphismAsMorphismFromDinstinguished- ObjectToHomomorphismStructure

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 ⇒ SolveLinearSystemInAbCategory

Applications CAP basic operations

HomomorphismStructureOnObjects HomomorphismStructureOnMorphisms DistinguishedObjectOfHomomorphismStructure InterpretMorphismAsMorphismFromDinstinguished- ObjectToHomomorphismStructure InterpretMorphismFromDinstinguished- ObjectToHomomorphismStructureAsMorphism

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 Applications CAP basic operations

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 39 / 65 Part III

Stable categories and homotopy categories in CAP

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 40 / 65 Hence, morphisms that factor through a projective object should be treated as zero morphisms, and ... any two morphisms whose subtraction factors through a projective object should be treated as identical morphisms.

Definition Let R be a ring and R-mod its module category. We define the stable module category of R, denoted by R-mod as follows: 1 Obj(R-mod) := Obj(R-mod), 2 For a, b ∈ R-mod, we define

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Stabilisation of a category by its projective objects

MOTTO: We want to identify projective objects with the zero object.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 any two morphisms whose subtraction factors through a projective object should be treated as identical morphisms.

Definition Let R be a ring and R-mod its module category. We define the stable module category of R, denoted by R-mod as follows: 1 Obj(R-mod) := Obj(R-mod), 2 For a, b ∈ R-mod, we define

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Stabilisation of a category by its projective objects

MOTTO: We want to identify projective objects with the zero object. Hence, morphisms that factor through a projective object should be treated as zero morphisms, and ...

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 Definition Let R be a ring and R-mod its module category. We define the stable module category of R, denoted by R-mod as follows: 1 Obj(R-mod) := Obj(R-mod), 2 For a, b ∈ R-mod, we define

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Stabilisation of a category by its projective objects

MOTTO: We want to identify projective objects with the zero object. Hence, morphisms that factor through a projective object should be treated as zero morphisms, and ... any two morphisms whose subtraction factors through a projective object should be treated as identical morphisms.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 1 Obj(R-mod) := Obj(R-mod), 2 For a, b ∈ R-mod, we deﬁne

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Stabilisation of a category by its projective objects

Deﬁnition Let R be a ring and R-mod its module category. We deﬁne the stable module category of R, denoted by R-mod as follows:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 2 For a, b ∈ R-mod, we deﬁne

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Stabilisation of a category by its projective objects

Deﬁnition Let R be a ring and R-mod its module category. We deﬁne the stable module category of R, denoted by R-mod as follows: 1 Obj(R-mod) := Obj(R-mod),

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 Stabilisation of a category by its projective objects

Definition Let R be a ring and R-mod its module category. We define the stable module category of R, denoted by R-mod as follows: 1 Obj(R-mod) := Obj(R-mod), 2 For a, b ∈ R-mod, we define

HomR-mod(a, b) := HomR-mod(a, b)/ ∼,

where ϕ ∼ ψ if ϕ − ψ factors through a projective object.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 41 / 65 it is enough to decide liftability of ϕ along πb.

∃ ∃ P ∃ ∃

Pb πb

P is projective object and the diagram commutes. Because any morphism from a projective module is liftable to any epimorphism,

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 it is enough to decide liftability of ϕ along πb.

∃ ∃ P ∃ ∃

Pb πb

P is projective object and the diagram commutes. Because any morphism from a projective module is liftable to any epimorphism,

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then a

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 it is enough to decide liftability of ϕ along πb.

∃ ∃

Pb πb

Because any morphism from a projective module is liftable to any epimorphism,

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then a ∃

P ϕ ∃

P is projective object and the diagram commutes.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 it is enough to decide liftability of ϕ along πb.

∃ ∃

Because any morphism from a projective module is liftable to any epimorphism,

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then a ∃

P ϕ ∃

Pb b πb

P is projective object and the diagram commutes.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 ∃

it is enough to decide liftability of ϕ along πb.

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then a ∃

P ϕ ∃ ∃

Pb b πb

P is projective object and the diagram commutes. Because any morphism from a projective module is liftable to any epimorphism,

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 ∃

P ∃ ∃

Stabilisation of a category by its projective objects

Suppose ϕ : a → b is an R-homomorphism that factors through a projective object. Furthermore, let πb : Pb b be an epimorphism from some projective object Pb. Then a ∃ ϕ

Pb b πb

P is projective object and the diagram commutes. Because any morphism from a projective module is liftable to any epimorphism, it is enough to decide liftability of ϕ along πb.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 42 / 65 IsProjective b is b projective?

SomeProjectiveObject b Pb

EpimorphismFrom- b πb : Pb b SomeProjectiveObject

ProjectiveLift (α : P → b, β : a b) δ : P → a

Categories with enough projective objects in CAP

A category A is called computable with enough projectives if A is computable and is equiped with following basic operations:

Basic operation Input Output

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 43 / 65 SomeProjectiveObject b Pb

EpimorphismFrom- b πb : Pb b SomeProjectiveObject

ProjectiveLift (α : P → b, β : a b) δ : P → a

Categories with enough projective objects in CAP

A category A is called computable with enough projectives if A is computable and is equiped with following basic operations:

Basic operation Input Output IsProjective b is b projective?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 43 / 65 EpimorphismFrom- b πb : Pb b SomeProjectiveObject

ProjectiveLift (α : P → b, β : a b) δ : P → a

Categories with enough projective objects in CAP

A category A is called computable with enough projectives if A is computable and is equiped with following basic operations:

Basic operation Input Output IsProjective b is b projective?

SomeProjectiveObject b Pb

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 43 / 65 ProjectiveLift (α : P → b, β : a b) δ : P → a

Categories with enough projective objects in CAP

A category A is called computable with enough projectives if A is computable and is equiped with following basic operations:

Basic operation Input Output IsProjective b is b projective?

SomeProjectiveObject b Pb

EpimorphismFrom- b πb : Pb b SomeProjectiveObject

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 43 / 65 Categories with enough projective objects in CAP

A category A is called computable with enough projectives if A is computable and is equiped with following basic operations:

Basic operation Input Output IsProjective b is b projective?

SomeProjectiveObject b Pb

EpimorphismFrom- b πb : Pb b SomeProjectiveObject

ProjectiveLift (α : P → b, β : a b) δ : P → a

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 43 / 65 a ?∃ ϕ

Pb b πb

The following code in CAP checks whether ϕ factors through a projective: gap> b := Range( phi ); gap> pi_b := EpimorphismFromSomeProjectiveObject( b ); gap> IsLiftable( phi, pi_b );

Stabilisation of a category by its projective objects

Back to our R-homomorphism ϕ : a → b.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 44 / 65 Stabilisation of a category by its projective objects

Back to our R-homomorphism ϕ : a → b. a ?∃ ϕ

Pb b πb

The following code in CAP checks whether ϕ factors through a projective: gap> b := Range( phi ); gap> pi_b := EpimorphismFromSomeProjectiveObject( b ); gap> IsLiftable( phi, pi_b );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 44 / 65 1 For any object a in A, there exists a morphism `a : La → a, for some object La ∈ LA 2 For any morphism ϕ : a → b, there is a lifting morphism Lϕ : La → Lb such that the following diagram commutes:

`a La a ϕ Lϕ

Lb b. `b

Lifting objects as an abstraction of projective objects

Deﬁnition

Let A be an additive category. A system of lifting objects LA in A is a distinguished class of objects with the following properties:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 45 / 65 2 For any morphism ϕ : a → b, there is a lifting morphism Lϕ : La → Lb such that the following diagram commutes:

`a La a ϕ Lϕ

Lb b. `b

Lifting objects as an abstraction of projective objects

Deﬁnition

Let A be an additive category. A system of lifting objects LA in A is a distinguished class of objects with the following properties:

1 For any object a in A, there exists a morphism `a : La → a, for some object La ∈ LA

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 45 / 65 Lifting objects as an abstraction of projective objects

Deﬁnition

Let A be an additive category. A system of lifting objects LA in A is a distinguished class of objects with the following properties:

1 For any object a in A, there exists a morphism `a : La → a, for some object La ∈ LA 2 For any morphism ϕ : a → b, there is a lifting morphism Lϕ : La → Lb such that the following diagram commutes:

`a La a ϕ Lϕ

Lb b. `b

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 45 / 65 Projective objects deﬁne a lifting system

For any additive category with enough projectives we have a lifting system deﬁned by:

`a := πa La := Pa a ϕ Lϕ := ProjectiveLift(πaϕ, πb)

Lb := Pb b `b := πb

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 46 / 65 We deﬁne the stable category of A by LA by

StabLA (A) := A/ILA , i.e., 1 The object class is same as that of A. 2 For two objects a, b ∈ A it is

Hom (a, b) := HomA(a, b)/IL (a, b), StabLA (A) A

where ILA (a, b) := ILA ∩ HomA(a, b).

Stable categories by a lifting system

Let A be an additive category.

If LA is a system of lifting objects for A, then

ILA = {ϕ : a → b| ϕ factors through `b}

is a two-sided ideal of morphisms in A.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 47 / 65 i.e., 1 The object class is same as that of A. 2 For two objects a, b ∈ A it is

Hom (a, b) := HomA(a, b)/IL (a, b), StabLA (A) A

where ILA (a, b) := ILA ∩ HomA(a, b).

Stable categories by a lifting system

Let A be an additive category.

If LA is a system of lifting objects for A, then

ILA = {ϕ : a → b| ϕ factors through `b}

is a two-sided ideal of morphisms in A.

We deﬁne the stable category of A by LA by

StabLA (A) := A/ILA ,

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 47 / 65 2 For two objects a, b ∈ A it is

Hom (a, b) := HomA(a, b)/IL (a, b), StabLA (A) A

where ILA (a, b) := ILA ∩ HomA(a, b).

Stable categories by a lifting system

Let A be an additive category.

If LA is a system of lifting objects for A, then

ILA = {ϕ : a → b| ϕ factors through `b}

is a two-sided ideal of morphisms in A.

We deﬁne the stable category of A by LA by

StabLA (A) := A/ILA , i.e., 1 The object class is same as that of A.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 47 / 65 Stable categories by a lifting system

Let A be an additive category.

If LA is a system of lifting objects for A, then

ILA = {ϕ : a → b| ϕ factors through `b}

is a two-sided ideal of morphisms in A.

We deﬁne the stable category of A by LA by

StabLA (A) := A/ILA , i.e., 1 The object class is same as that of A. 2 For two objects a, b ∈ A it is

Hom (a, b) := HomA(a, b)/IL (a, b), StabLA (A) A

where ILA (a, b) := ILA ∩ HomA(a, b).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 47 / 65 IsLiftingObject b b ∈ LA?

LiftingObject b Lb

MorphismFrom- b `b : Lb → b LiftingObject

LiftingMorphism ϕ : a → b Lϕ : La → Lb

IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 LiftingObject b Lb

MorphismFrom- b `b : Lb → b LiftingObject

LiftingMorphism ϕ : a → b Lϕ : La → Lb

IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

IsLiftingObject b b ∈ LA?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 MorphismFrom- b `b : Lb → b LiftingObject

LiftingMorphism ϕ : a → b Lϕ : La → Lb

IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

IsLiftingObject b b ∈ LA?

LiftingObject b Lb

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 LiftingMorphism ϕ : a → b Lϕ : La → Lb

IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

IsLiftingObject b b ∈ LA?

LiftingObject b Lb

MorphismFrom- b `b : Lb → b LiftingObject

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

IsLiftingObject b b ∈ LA?

LiftingObject b Lb

MorphismFrom- b `b : Lb → b LiftingObject

LiftingMorphism ϕ : a → b Lϕ : La → Lb

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 Categories with lifting objects in CAP

A category A is called computable with a system of lifting objects LA if A is computable and is equipped with the following basic operations:

Basic operation Input Output

IsLiftingObject b b ∈ LA?

LiftingObject b Lb

MorphismFrom- b `b : Lb → b LiftingObject

LiftingMorphism ϕ : a → b Lϕ : La → Lb

IsLiftableThroughLiftingObject ϕ : a → b ϕ ∈ ILA (a, b)?

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 48 / 65 If lifts are computable in A, then StabLA (A) is computable additive.

The canonical projection functor A → StabLA (A) is additive. If A has a D-homomorphism structure such that D is abelian with

a projective distinguished object, then StabLA (A) has a D-homomorphism structure.

Remark All axioms and statements here can be dualized. The dual notation of a lifting object will be called a colifting object.

Computability of stable categories

Theorem Let A be a computable additive category equipped with a lifting system LA.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 49 / 65 The canonical projection functor A → StabLA (A) is additive. If A has a D-homomorphism structure such that D is abelian with

a projective distinguished object, then StabLA (A) has a D-homomorphism structure.

Remark All axioms and statements here can be dualized. The dual notation of a lifting object will be called a colifting object.

Computability of stable categories

Theorem Let A be a computable additive category equipped with a lifting system LA.

If lifts are computable in A, then StabLA (A) is computable additive.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 49 / 65 If A has a D-homomorphism structure such that D is abelian with

a projective distinguished object, then StabLA (A) has a D-homomorphism structure.

Remark All axioms and statements here can be dualized. The dual notation of a lifting object will be called a colifting object.

Computability of stable categories

Theorem Let A be a computable additive category equipped with a lifting system LA.

If lifts are computable in A, then StabLA (A) is computable additive.

The canonical projection functor A → StabLA (A) is additive.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 49 / 65 Remark All axioms and statements here can be dualized. The dual notation of a lifting object will be called a colifting object.

Computability of stable categories

Theorem Let A be a computable additive category equipped with a lifting system LA.

If lifts are computable in A, then StabLA (A) is computable additive.

The canonical projection functor A → StabLA (A) is additive. If A has a D-homomorphism structure such that D is abelian with

a projective distinguished object, then StabLA (A) has a D-homomorphism structure.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 49 / 65 Computability of stable categories

Theorem Let A be a computable additive category equipped with a lifting system LA.

If lifts are computable in A, then StabLA (A) is computable additive.

The canonical projection functor A → StabLA (A) is additive. If A has a D-homomorphism structure such that D is abelian with

a projective distinguished object, then StabLA (A) has a D-homomorphism structure.

Remark All axioms and statements here can be dualized. The dual notation of a lifting object will be called a colifting object.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 49 / 65 fpres ' ab, Z

Z, gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ : FinalizeCategory := false ); Category of left presentations of Z gap> AddMorphismFromLiftingObject( ab, > EpimorphismFromSomeProjectiveObject ); gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 fpres ' ab, Z

gap> ab := LeftPresentations( ZZ : FinalizeCategory := false ); Category of left presentations of Z gap> AddMorphismFromLiftingObject( ab, > EpimorphismFromSomeProjectiveObject ); gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

Z, gap> ZZ := HomalgRingOfIntegers( ); Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 ' ab,

gap> AddMorphismFromLiftingObject( ab, > EpimorphismFromSomeProjectiveObject ); gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

, fpres Z Z gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ : FinalizeCategory := false ); Category of left presentations of Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> AddMorphismFromLiftingObject( ab, > EpimorphismFromSomeProjectiveObject ); gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

, fpres ' ab, Z Z gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ : FinalizeCategory := false ); Category of left presentations of Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> CanCompute( ab, "LiftingMorphism" ); true gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

, fpres ' ab, Z Z gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ : FinalizeCategory := false ); Category of left presentations of Z gap> AddMorphismFromLiftingObject( ab, > EpimorphismFromSomeProjectiveObject ); gap> Finalize( ab ); true gap> InfoOfInstalledOperationsOfCategory( ab ); 44 primitive operations were used to derive 196 basic ones for this ab Abelian category with enough projectives category

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> IsAbelianCategoryWithEnoughProjectives( ab ); true

CAP demo: ﬁnitely-generated abelian groups ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 CAP demo: ﬁnitely-generated abelian groups ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 50 / 65 gap> stable_functor := CanonicalProjectionFunctor( stable_ab ); Canonical projection functor ... gap> m := HomalgMatrix( "[ [ 2, 3, 4 ], [ 3, 4, 3 ] ]", ZZ );; gap> a := AsLeftPresentation( m ); gap> stable_a := ApplyFunctor( stable_functor, a ); gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

, fpres ' ab, Stabprojs(fpres ) Z Z Z gap> stable_ab := StableCategoryByLiftingStructure( ab ); The stable category of Category of left presentations of Z ...

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> m := HomalgMatrix( "[ [ 2, 3, 4 ], [ 3, 4, 3 ] ]", ZZ );; gap> a := AsLeftPresentation( m ); gap> stable_a := ApplyFunctor( stable_functor, a ); gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> a := AsLeftPresentation( m ); gap> stable_a := ApplyFunctor( stable_functor, a ); gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> stable_a := ApplyFunctor( stable_functor, a ); gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 gap> InfoOfInstalledOperationsOfCategory( stable_ab ); 20 primitive operations were used to derive 73 basic ones ...

CAP demo: stable category of ab by lifting objects

, fpres ' ab, Stabprojs(fpres ) Z Z Z gap> stable_ab := StableCategoryByLiftingStructure( ab ); The stable category of Category of left presentations of Z ... gap> stable_functor := CanonicalProjectionFunctor( stable_ab ); Canonical projection functor ... gap> m := HomalgMatrix( "[ [ 2, 3, 4 ], [ 3, 4, 3 ] ]", ZZ );; gap> a := AsLeftPresentation( m ); gap> stable_a := ApplyFunctor( stable_functor, a ); gap> List( [ a, stable_a ], IsZeroForObjects ); [ false, true ] gap> SmithNormalFormIntegerMat( [ [ 2, 3, 4 ], [ 3, 4, 3 ] ] ); [ [ 1, 0, 0 ], [ 0, 1, 0 ] ]

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 CAP demo: stable category of ab by lifting objects

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 51 / 65 The same holds for the exterior algebra E of a Q-vector space. The same holds for the category of ﬁnite representations freps(Q, I) of an acyclic quiver Q with relations given by ideal I.

More examples

We can compute the example of the previous slide for any commutative computable ring R.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 52 / 65 The same holds for the category of ﬁnite representations freps(Q, I) of an acyclic quiver Q with relations given by ideal I.

More examples

We can compute the example of the previous slide for any commutative computable ring R. The same holds for the exterior algebra E of a Q-vector space.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 52 / 65 More examples

We can compute the example of the previous slide for any commutative computable ring R. The same holds for the exterior algebra E of a Q-vector space. The same holds for the category of ﬁnite representations freps(Q, I) of an acyclic quiver Q with relations given by ideal I.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 52 / 65 and let (hi : Ai → Bi+1)i∈Z be a family of morphisms in A:

A A di di+1 A• : ··· Ai−1 Ai Ai+1 ···

hi−1 hi

B• : ··· Bi−1 Bi Bi+1 ··· B B di di+1

B A Then the morphisms (ϕi := hi di+1 + di hi−1 : Ai → Bi )i∈Z deﬁne a chain morphism ϕ• : A• → B•. Such chain morphisms are called null-homotopic.

Null-homotopic chain morphisms

Let A be an additive category and A•, B• be objects in its category of b complexes Ch•(A)

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 53 / 65 B A Then the morphisms (ϕi := hi di+1 + di hi−1 : Ai → Bi )i∈Z deﬁne a chain morphism ϕ• : A• → B•. Such chain morphisms are called null-homotopic.

Null-homotopic chain morphisms

Let A be an additive category and A•, B• be objects in its category of b complexes Ch•(A) and let (hi : Ai → Bi+1)i∈Z be a family of morphisms in A:

A A di di+1 A• : ··· Ai−1 Ai Ai+1 ···

hi−1 hi

B• : ··· Bi−1 Bi Bi+1 ··· B B di di+1

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 53 / 65 Such chain morphisms are called null-homotopic.

Null-homotopic chain morphisms

Let A be an additive category and A•, B• be objects in its category of b complexes Ch•(A) and let (hi : Ai → Bi+1)i∈Z be a family of morphisms in A:

A A di di+1 A• : ··· Ai−1 Ai Ai+1 ···

ϕ ϕ i−1 hi−1 ϕi hi i+1

B• : ··· Bi−1 Bi Bi+1 ··· B B di di+1

B A Then the morphisms (ϕi := hi di+1 + di hi−1 : Ai → Bi )i∈Z deﬁne a chain morphism ϕ• : A• → B•.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 53 / 65 Null-homotopic chain morphisms

Let A be an additive category and A•, B• be objects in its category of b complexes Ch•(A) and let (hi : Ai → Bi+1)i∈Z be a family of morphisms in A:

A A di di+1 A• : ··· Ai−1 Ai Ai+1 ···

ϕ ϕ i−1 hi−1 ϕi hi i+1

B• : ··· Bi−1 Bi Bi+1 ··· B B di di+1

B A Then the morphisms (ϕi := hi di+1 + di hi−1 : Ai → Bi )i∈Z deﬁne a chain morphism ϕ• : A• → B•. Such chain morphisms are called null-homotopic.

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 53 / 65 Cone(ϕ•) where di for i ∈ Z is given by the following matrix:

A di−1 −ϕi−1 B . 0 di

Moreover, we have the obvious natural injection of B• in Cone(ϕ•),

usually denoted by ιϕ• .

Mapping cone

For a chain morphism ϕ• : A• → B• we deﬁne the cone of ϕ•, denoted by Cone(ϕ•), to be the following complex:

Cone(ϕ•) di ··· Ai−2 ⊕ Bi−1 Ai−1 ⊕ Bi ···

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 54 / 65 Moreover, we have the obvious natural injection of B• in Cone(ϕ•),

usually denoted by ιϕ• .

Mapping cone

For a chain morphism ϕ• : A• → B• we deﬁne the cone of ϕ•, denoted by Cone(ϕ•), to be the following complex:

Cone(ϕ•) di ··· Ai−2 ⊕ Bi−1 Ai−1 ⊕ Bi ···

Cone(ϕ•) where di for i ∈ Z is given by the following matrix:

A di−1 −ϕi−1 B . 0 di

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 54 / 65 Mapping cone

For a chain morphism ϕ• : A• → B• we deﬁne the cone of ϕ•, denoted by Cone(ϕ•), to be the following complex:

Cone(ϕ•) di ··· Ai−2 ⊕ Bi−1 Ai−1 ⊕ Bi ···

Cone(ϕ•) where di for i ∈ Z is given by the following matrix:

A di−1 −ϕi−1 B . 0 di

Moreover, we have the obvious natural injection of B• in Cone(ϕ•), usually denoted by ιϕ• .

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 54 / 65 gap> A := Source( phi ); id_A := IdentityMorphism( A ); gap> iota := NaturalInjectionInMappingCone( id_A ); gap> IsColiftable( iota, phi ); or gap> IsNullHomotopic( phi );

Mapping cone and null-homotopic morphisms

A chain morphism ϕ• : A• → B• is null-homotopic iff ιid(A•) is coliftable along ϕ•, i.e., there is δ such that the following diagram commutes:

ϕ• A• B•

ι idA• δ

Cone(idA• )

The following code in CAP checks whether ϕ is null-homotopic:

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 55 / 65 gap> iota := NaturalInjectionInMappingCone( id_A ); gap> IsColiftable( iota, phi ); or gap> IsNullHomotopic( phi );

Mapping cone and null-homotopic morphisms

A chain morphism ϕ• : A• → B• is null-homotopic iff ιid(A•) is coliftable along ϕ•, i.e., there is δ such that the following diagram commutes:

ϕ• A• B•

ι idA• δ

Cone(idA• )

The following code in CAP checks whether ϕ is null-homotopic: gap> A := Source( phi ); id_A := IdentityMorphism( A );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 55 / 65 gap> IsColiftable( iota, phi ); or gap> IsNullHomotopic( phi );

Mapping cone and null-homotopic morphisms

A chain morphism ϕ• : A• → B• is null-homotopic iff ιid(A•) is coliftable along ϕ•, i.e., there is δ such that the following diagram commutes:

ϕ• A• B•

ι idA• δ

Cone(idA• )

The following code in CAP checks whether ϕ is null-homotopic: gap> A := Source( phi ); id_A := IdentityMorphism( A ); gap> iota := NaturalInjectionInMappingCone( id_A );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 55 / 65 or gap> IsNullHomotopic( phi );

Mapping cone and null-homotopic morphisms

A chain morphism ϕ• : A• → B• is null-homotopic iff ιid(A•) is coliftable along ϕ•, i.e., there is δ such that the following diagram commutes:

ϕ• A• B•

ι idA• δ

Cone(idA• )

The following code in CAP checks whether ϕ is null-homotopic: gap> A := Source( phi ); id_A := IdentityMorphism( A ); gap> iota := NaturalInjectionInMappingCone( id_A ); gap> IsColiftable( iota, phi );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 55 / 65 Mapping cone and null-homotopic morphisms

A chain morphism ϕ• : A• → B• is null-homotopic iff ιid(A•) is coliftable along ϕ•, i.e., there is δ such that the following diagram commutes:

ϕ• A• B•

ι idA• δ

Cone(idA• )

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 55 / 65 Deﬁnition Let A be an additive category, then the bounded homotopy category of A, denoted by Hob(A) is deﬁned by

b b Ho (A) := StabC(Ch•(A)).

Theorem Let A be a computable additive category with D-homomorphism structure such that D is abelian with a projective distinguished object. Then Hob(A) is computable additive and has a D-homomorphism structure.

b System of colifting objects for Ch• (A)

Lemma The class C of split exact complexes is a system of colifting objects for Chb(A), explicitly C := Cone(id ) and c := ι : A → C . • A• A• A• idA• • A•

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 56 / 65 Theorem Let A be a computable additive category with D-homomorphism structure such that D is abelian with a projective distinguished object. Then Hob(A) is computable additive and has a D-homomorphism structure.

The bounded homotopy category

Lemma The class C of split exact complexes is a system of colifting objects for Chb(A), explicitly C := Cone(id ) and c := ι : A → C . • A• A• A• idA• • A•

Deﬁnition Let A be an additive category, then the bounded homotopy category of A, denoted by Hob(A) is deﬁned by

b b Ho (A) := StabC(Ch•(A)).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 56 / 65 Computability of bounded homotopy categories

Lemma The class C of split exact complexes is a system of colifting objects for Chb(A), explicitly C := Cone(id ) and c := ι : A → C . • A• A• A• idA• • A•

Deﬁnition Let A be an additive category, then the bounded homotopy category of A, denoted by Hob(A) is deﬁned by

b b Ho (A) := StabC(Ch•(A)).

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 56 / 65 gap> ab := LeftPresentations( ZZ ); Category of left presentations of Z gap> H := HomotopyCategory( ab ); Homotopy category of Category of left presentations of Z gap> InfoOfInstalledOperationsOfCategory( H ); 32 primitive operations were used to derive 92 basic ones for this ab additive category gap> F := CanonicalProjectionFunctor( H ); Canonical projection functor from Chain complexes category over Category of left presentations of Z in Homotopy category of Category of left presentations of Z

CAP demo: the bounded homotopy category of ab

Let us go back to our ﬁrst example gap> ZZ := HomalgRingOfIntegers( ); Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 57 / 65 gap> H := HomotopyCategory( ab ); Homotopy category of Category of left presentations of Z gap> InfoOfInstalledOperationsOfCategory( H ); 32 primitive operations were used to derive 92 basic ones for this ab additive category gap> F := CanonicalProjectionFunctor( H ); Canonical projection functor from Chain complexes category over Category of left presentations of Z in Homotopy category of Category of left presentations of Z

CAP demo: the bounded homotopy category of ab

Let us go back to our ﬁrst example gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ ); Category of left presentations of Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 57 / 65 gap> InfoOfInstalledOperationsOfCategory( H ); 32 primitive operations were used to derive 92 basic ones for this ab additive category gap> F := CanonicalProjectionFunctor( H ); Canonical projection functor from Chain complexes category over Category of left presentations of Z in Homotopy category of Category of left presentations of Z

CAP demo: the bounded homotopy category of ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 57 / 65 gap> F := CanonicalProjectionFunctor( H ); Canonical projection functor from Chain complexes category over Category of left presentations of Z in Homotopy category of Category of left presentations of Z

CAP demo: the bounded homotopy category of ab

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 57 / 65 CAP demo: the bounded homotopy category of ab

Let us go back to our ﬁrst example gap> ZZ := HomalgRingOfIntegers( ); Z gap> ab := LeftPresentations( ZZ ); Category of left presentations of Z gap> H := HomotopyCategory( ab ); Homotopy category of Category of left presentations of Z gap> InfoOfInstalledOperationsOfCategory( H ); 32 primitive operations were used to derive 92 basic ones for this ab additive category gap> F := CanonicalProjectionFunctor( H ); Canonical projection functor from Chain complexes category over Category of left presentations of Z in Homotopy category of Category of left presentations of Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 57 / 65 gap> C_1 := FreeLeftPresentation( 1, ZZ );; gap> C_0 := DirectSum( C_1, C_1 );; gap> m := HomalgMatrix( [[1,3]], ZZ );; gap> dC_1 := PresentationMorphism( C_1, m, C_0 );; gap> C := ChainComplex( [ dC_1 ], 1 ); gap> D_1 := C_0; gap> D_0 := AsLeftPresentation( HomalgMatrix( [[0,2]], ZZ ) );; gap> m := HomalgMatrix( [[2,3],[5,1]], ZZ );; gap> dD_1 := PresentationMorphism( D_1, m, D_0 );; gap> D := ChainComplex( [ dD_1 ], 1 );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 58 / 65 gap> m := HomalgMatrix( [[1,3]], ZZ );; gap> dC_1 := PresentationMorphism( C_1, m, C_0 );; gap> C := ChainComplex( [ dC_1 ], 1 ); gap> D_1 := C_0; gap> D_0 := AsLeftPresentation( HomalgMatrix( [[0,2]], ZZ ) );; gap> m := HomalgMatrix( [[2,3],[5,1]], ZZ );; gap> dD_1 := PresentationMorphism( D_1, m, D_0 );; gap> D := ChainComplex( [ dD_1 ], 1 );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> C_1 := FreeLeftPresentation( 1, ZZ );; gap> C_0 := DirectSum( C_1, C_1 );;

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 58 / 65 gap> D_1 := C_0; gap> D_0 := AsLeftPresentation( HomalgMatrix( [[0,2]], ZZ ) );; gap> m := HomalgMatrix( [[2,3],[5,1]], ZZ );; gap> dD_1 := PresentationMorphism( D_1, m, D_0 );; gap> D := ChainComplex( [ dD_1 ], 1 );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 58 / 65 gap> m := HomalgMatrix( [[2,3],[5,1]], ZZ );; gap> dD_1 := PresentationMorphism( D_1, m, D_0 );; gap> D := ChainComplex( [ dD_1 ], 1 );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 58 / 65 CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> C_1 := FreeLeftPresentation( 1, ZZ );; gap> C_0 := DirectSum( C_1, C_1 );; gap> m := HomalgMatrix( [[1,3]], ZZ );; gap> dC_1 := PresentationMorphism( C_1, m, C_0 );; gap> C := ChainComplex( [ dC_1 ], 1 ); gap> D_1 := C_0; gap> D_0 := AsLeftPresentation( HomalgMatrix( [[0,2]], ZZ ) );; gap> m := HomalgMatrix( [[2,3],[5,1]], ZZ );; gap> dD_1 := PresentationMorphism( D_1, m, D_0 );; gap> D := ChainComplex( [ dD_1 ], 1 );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 58 / 65 gap> m := HomalgMatrix( [[15,7]], ZZ );; gap> phi_1 := PresentationMorphism( C_1, m, D_1 ); gap> phi := ChainMorphism( C, D, [ phi_0, phi_1 ], 0 ); gap> F_phi := ApplyFunctor( F, phi ); gap> IsZero( F_phi ); true gap> h_morphisms := HomotopyMorphisms( F_phi );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> m := HomalgMatrix( [[26,13],[13,13]], ZZ );; gap> phi_0 := PresentationMorphism( C_0, m, D_0 );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 59 / 65 gap> phi := ChainMorphism( C, D, [ phi_0, phi_1 ], 0 ); gap> F_phi := ApplyFunctor( F, phi ); gap> IsZero( F_phi ); true gap> h_morphisms := HomotopyMorphisms( F_phi );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> m := HomalgMatrix( [[26,13],[13,13]], ZZ );; gap> phi_0 := PresentationMorphism( C_0, m, D_0 ); gap> m := HomalgMatrix( [[15,7]], ZZ );; gap> phi_1 := PresentationMorphism( C_1, m, D_1 );

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 59 / 65 gap> h_morphisms := HomotopyMorphisms( F_phi );

CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> m := HomalgMatrix( [[26,13],[13,13]], ZZ );; gap> phi_0 := PresentationMorphism( C_0, m, D_0 ); gap> m := HomalgMatrix( [[15,7]], ZZ );; gap> phi_1 := PresentationMorphism( C_1, m, D_1 ); gap> phi := ChainMorphism( C, D, [ phi_0, phi_1 ], 0 ); gap> F_phi := ApplyFunctor( F, phi ); gap> IsZero( F_phi ); true

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 59 / 65 CAP demo: the bounded homotopy category of ab

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 59 / 65 gap> h0 := h_morphisms[ 0 ]; gap> Display( h0 ); [ [ 633, -248 ], [ -206, 85 ] ]

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 60 / 65 gap> Display( h0 ); [ [ 633, -248 ], [ -206, 85 ] ]

A morphism in Category of left presentations of Z gap> IsCongruentForMorphisms( PreCompose( C^1, h0 ), phi[1] ); true gap> IsCongruentForMorphisms( PreCompose( h0, D^1 ), phi[0] ); true

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> h0 := h_morphisms[ 0 ];

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 60 / 65 gap> IsCongruentForMorphisms( PreCompose( C^1, h0 ), phi[1] ); true gap> IsCongruentForMorphisms( PreCompose( h0, D^1 ), phi[0] ); true

1 3 × × C : 0 Z1 2 Z1 1 0 26 13 15 7 13 13 2 3 1×2 5 1 Z 1×2 D : 0 h(0 2)i Z 0

gap> h0 := h_morphisms[ 0 ]; gap> Display( h0 ); [ [ 633, -248 ], [ -206, 85 ] ]

A morphism in Category of left presentations of Z

Posur, Saleh, Zickgraf (Siegen) CAP July 15, 2019 60 / 65 gap> IsCongruentForMorphisms( PreCompose( h0, D^1 ), phi[0] ); true