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2 Lifting and Colifting systems

2.1 Systems of lifting objects

Let \CC be an additive category. A system of lifting objects in \CC consists of the following data:

• A distinguished class \LL of objects.

• Every object A is equipped with a distinguished object L_A in \LL and a distinguished morphism \ell_A:L_A \to A. Furthermore, if A belongs to \LL, then \ell_A is a split-epimorphism.

• For every morphism \alpha:A\to B, there exists a morphism L_{\alpha} with \comp{L_\alpha}{\ell_B} \sim \comp{\ell_A}{\alpha}, i.e., we get a commutative diagram:

2.1-1 IsLiftingObject
 ‣ IsLiftingObject( A ) ( property )

Returns: true or false

The argument is an object A. The output is whether or not A belongs to \LL.

2.1-2 LiftingObject
 ‣ LiftingObject( A ) ( attribute )

Returns: an object in \LL

The argument is an object A. The output is an object L_A in \LL.

2.1-3 MorphismFromLiftingObject
 ‣ MorphismFromLiftingObject( A ) ( attribute )

Returns: a morphism L_A \to A

The argument is an object A. The output is the distinguished morphism \ell_A:L_A \to A where L_A=\mathrm{LiftingObject}(A).

2.1-4 MorphismFromLiftingObjectWithGivenLiftingObject
 ‣ MorphismFromLiftingObjectWithGivenLiftingObject( A, L_A ) ( operation )

Returns: a morphism L_A \to A

The arguments are two objects A and L_A=\mathrm{LiftingObject}(A). The output is the distinguished morphism \ell_A:L_A \to A.

2.1-5 SectionOfMorphismFromLiftingObjectWithGivenLiftingObject
 ‣ SectionOfMorphismFromLiftingObjectWithGivenLiftingObject( A, L_A ) ( operation )

Returns: a morphism A \to L_A

The argument is a lifting object A in \LL and L_A=\mathrm{LiftingObject}(A). The output is a section morphism s_A:A \to L_A of \ell_A = \mathrm{MorphismFromLiftingObjectWithGivenLiftingObject}(A,L_A).

2.1-6 SectionOfMorphismFromLiftingObject
 ‣ SectionOfMorphismFromLiftingObject( A ) ( attribute )

Returns: a morphism A \to L_A

The argument is a lifting object A in \LL. The output is a section morphism s_A:A \to L_A of \ell_A = \mathrm{MorphismFromLiftingObject}(A).

2.1-7 LiftingMorphismWithGivenLiftingObjects
 ‣ LiftingMorphismWithGivenLiftingObjects( L_A, alpha, L_B ) ( operation )

Returns: a morphism L_A \to L_B

The arguments are an object L_A=\mathrm{LiftingObject}(A), a morphism \alpha:A \to B and an object L_B=\mathrm{LiftingObject}(B). The output is a morphism L_{\alpha}:L_A \to L_B with \comp{L_\alpha}{\ell_B} \sim \comp{\ell_A}{\alpha} where \ell_A=\mathrm{MorphismFromLiftingObject}(A) and \ell_B=\mathrm{MorphismFromLiftingObject}(B).

2.1-8 LiftingMorphism
 ‣ LiftingMorphism( alpha ) ( attribute )

Returns: a morphism L_A \to L_B

The argument is a morphism \alpha:A \to B. The output is a morphism L_{\alpha}:L_A \to L_B with \comp{L_\alpha}{\ell_B} \sim \comp{\ell_A}{\alpha} where L_A=\mathrm{LiftingObject}(A), L_B=\mathrm{LiftingObject}(B), \ell_A=\mathrm{MorphismFromLiftingObject}(A) and \ell_B=\mathrm{MorphismFromLiftingObject}(B).

2.1-9 IsLiftableAlongMorphismFromLiftingObject
 ‣ IsLiftableAlongMorphismFromLiftingObject( alpha ) ( property )

Returns: true or false

The argument is a morphism \alpha:A \to B. The output is whether or not \alpha lifts along \ell_B: L_B \to B where \ell_B=\mathrm{MorphismFromLiftingObject}(B).

2.1-10 WitnessForBeingLiftableAlongMorphismFromLiftingObject
 ‣ WitnessForBeingLiftableAlongMorphismFromLiftingObject( alpha ) ( attribute )

Returns: a morphism \lambda:A \to L_B

The argument is a morphism \alpha:A \to B which lifts along \ell_B:L_B \to B, where \ell_B=\mathrm{MorphismFromLiftingObject}(B). The output is a lift morphism \lambda:A \to L_B of \alpha along \ell_B, i.e., \comp{\lambda}{\ell_B} \sim \alpha.

2.2 Systems of colifting objects

Let \CC be an additive category. A system of colifting objects in \CC consists of the following data:

• A distinguished class \QQ of objects.

• Every object A is equipped with a distinguished object Q_A in \QQ and a distinguished morphism q_A:A \to Q_A. Furthermore, if A belongs to \QQ, then q_A is a split-monomorphism.

• For every morphism \alpha:A\to B, there exists a morphism Q_{\alpha} with \comp{q_A}{Q_\alpha} \sim \comp{\alpha}{q_B}, i.e., we get a commutative diagram:

2.2-1 IsColiftingObject
 ‣ IsColiftingObject( A ) ( property )

Returns: true or false

The argument is an object A. The output is whether or not A belongs to \QQ.

2.2-2 ColiftingObject
 ‣ ColiftingObject( A ) ( attribute )

Returns: an object in \QQ

The argument is an object A. The output is an object Q_A in \QQ.

2.2-3 MorphismToColiftingObject
 ‣ MorphismToColiftingObject( A ) ( attribute )

Returns: a morphism A \to Q_A

The argument is an object A. The output is the distinguished morphism q_A:A \to Q_A where Q_A=\mathrm{ColiftingObject}(A).

2.2-4 MorphismToColiftingObjectWithGivenColiftingObject
 ‣ MorphismToColiftingObjectWithGivenColiftingObject( A, Q_A ) ( operation )

Returns: a morphism A \to Q_A

The arguments are two objects A and Q_A=\mathrm{ColiftingObject}(A). The output is the distinguished morphism q_A:A \to Q_A.

2.2-5 RetractionOfMorphismToColiftingObjectWithGivenColiftingObject
 ‣ RetractionOfMorphismToColiftingObjectWithGivenColiftingObject( A, Q_A ) ( operation )

Returns: a morphism Q_A \to A

The argument is a colifting object A in \QQ and Q_A=\mathrm{ColiftingObject}(A). The output is a retraction morphism r_A:Q_A \to A of q_A = \mathrm{MorphismToColiftingObjectWithGivenColiftingObject}(A,Q_A).

2.2-6 RetractionOfMorphismToColiftingObject
 ‣ RetractionOfMorphismToColiftingObject( A ) ( attribute )

Returns: a morphism Q_A \to A

The argument is a colifting object A in \QQ. The output is a retraction morphism r_A:Q_A \to A of q_A = \mathrm{MorphismToColiftingObject}(A).

2.2-7 ColiftingMorphismWithGivenColiftingObjects
 ‣ ColiftingMorphismWithGivenColiftingObjects( Q_A, alpha, Q_B ) ( operation )

Returns: a morphism Q_A \to Q_B

The arguments are an object Q_A=\mathrm{ColiftingObject}(A), a morphism \alpha:A \to B and an object Q_B=\mathrm{ColiftingObject}(B). The output is a morphism Q_{\alpha}:Q_A \to Q_B with \comp{q_A}{Q_\alpha} \sim \comp{\alpha}{q_B} where q_A=\mathrm{MorphismToColiftingObject}(A) and q_B=\mathrm{MorphismToColiftingObject}(B).

2.2-8 ColiftingMorphism
 ‣ ColiftingMorphism( alpha ) ( attribute )

Returns: a morphism Q_A \to Q_B

The argument is a morphism \alpha : A \to B. The output is a morphism Q_{\alpha}:Q_A \to Q_B with \comp{q_A}{Q_\alpha} \sim \comp{\alpha}{q_B} where Q_A=\mathrm{ColiftingObject}(A), Q_B=\mathrm{ColiftingObject}(B), q_A=\mathrm{MorphismToColiftingObject}(A) and q_B=\mathrm{MorphismToColiftingObject}(B).

2.2-9 IsColiftableAlongMorphismToColiftingObject
 ‣ IsColiftableAlongMorphismToColiftingObject( alpha ) ( property )

Returns: true or false

The argument is a morphism \alpha:A \to B. The output is whether or not \alpha colifts along q_A: A \to Q_A where q_A=\mathrm{MorphismToColiftingObject}(A).

2.2-10 WitnessForBeingColiftableAlongMorphismToColiftingObject
 ‣ WitnessForBeingColiftableAlongMorphismToColiftingObject( alpha ) ( attribute )

Returns: a morphism \lambda:A \to L_B

The argument is a morphism \alpha:A \to B which colifts along q_A:A \to Q_A, where q_A=\mathrm{MorphismToColiftingObject}(A). The output is a colift morphism \lambda:Q_A \to B of \alpha along q_A, i.e., \comp{q_A}{\lambda} \sim \alpha.

2.3 Examples for systems of colifting objects

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