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:
‣ IsLiftingObject( A ) | ( property ) |
Returns: true or false
The argument is an object A. The output is whether or not A belongs to \LL.
‣ LiftingObject( A ) | ( attribute ) |
Returns: an object in \LL
The argument is an object A. The output is an object L_A in \LL.
‣ 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).
‣ 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.
‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ 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.
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:
‣ IsColiftingObject( A ) | ( property ) |
Returns: true or false
The argument is an object A. The output is whether or not A belongs to \QQ.
‣ ColiftingObject( A ) | ( attribute ) |
Returns: an object in \QQ
The argument is an object A. The output is an object Q_A in \QQ.
‣ 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).
‣ 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.
‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ 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.
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