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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