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### 1 Exact and Frobenius Categories

#### 1.1 GAP categories

##### 1.1-1 IsCapCategoryShortSequence
 ‣ IsCapCategoryShortSequence( seq_obj ) ( filter )

Returns: true or false

The GAP category of short sequences.

##### 1.1-2 IsCapCategoryMorphismOfShortSequences
 ‣ IsCapCategoryMorphismOfShortSequences( seq_mor ) ( filter )

Returns: true or false

The GAP category of morphisms of short sequences.

##### 1.1-3 IsCapCategoryShortExactSequence
 ‣ IsCapCategoryShortExactSequence( seq_obj ) ( filter )

Returns: true or false

The GAP category of short exact sequences.

##### 1.1-4 IsCapCategoryConflation
 ‣ IsCapCategoryConflation( seq_obj ) ( filter )

Returns: true or false

The GAP category of conflations. If a short sequence is a conflation, then it is a short exact sequence.

#### 1.2 Exact categories operations

##### 1.2-1 IsExactCategory
 ‣ IsExactCategory( C ) ( property )

Returns: true or false

The input is a CAP category. The output is true if \CC is an exact category with respect to some class \EE of short exact sequences.

##### 1.2-2 IsInflation
 ‣ IsInflation( iota ) ( attribute )

Returns: a boolian

The argument is a morphism \iota:A\to B in \CC. The output is whether or not \iota is an inflation.

##### 1.2-3 IsDeflation
 ‣ IsDeflation( pi ) ( attribute )

Returns: a boolian

The argument is a morphism \pi:B\to C in \CC. The output is whether or not \pi is a deflation.

##### 1.2-4 IsConflationPair
 ‣ IsConflationPair( iota, pi ) ( operation )

Returns: a boolian

The argument is a pair of morphisms \iota:A\to B and \pi:B\to C. The output is whether or not the pair (\iota,\pi) defines a conflation.

##### 1.2-5 ExactCokernelObject
 ‣ ExactCokernelObject( iota ) ( attribute )

Returns: an object C

The argument is an inflation \iota:A\to B. The output is the cokernel object C of \iota.

##### 1.2-6 ExactCokernelProjection
 ‣ ExactCokernelProjection( iota ) ( attribute )

Returns: a deflation B\to C

The argument is an inflation \iota:A\to B. The output is a deflation \pi(\iota):B\to C with C=\mathrm{ExactCokernelObject}(\iota) such that the pair (\iota,\pi(\iota)) defines a conflation.

##### 1.2-7 ExactCokernelProjectionWithGivenExactCokernelObject
 ‣ ExactCokernelProjectionWithGivenExactCokernelObject( iota, C ) ( operation )

Returns: a deflation B\to C

The argument is an inflation \iota:A\to B and an object C=\mathrm{ExactCokernelObject}(\iota). The output is a deflation \pi(\iota):B\to C such that (\iota,\pi(\iota)) defines a conflation.

##### 1.2-8 ExactCokernelColift
 ‣ ExactCokernelColift( iota, tau ) ( operation )

Returns: a morphism C \to T

The arguments are an inflation \iota: A \rightarrow B and a test morphism \tau: B \rightarrow T satisfying \comp{\iota}{\tau} \sim 0. The output is the morphism \lambda: C \rightarrow T with C=\mathrm{ExactCokernelObject}(\iota) and \lambda is given by the universal property of the cokernel object, i.e., \comp{\pi(\iota)}{\lambda} \sim \tau where \pi(\iota) = \mathrm{ExactCokernelProjection}(\iota).

##### 1.2-9 ExactCokernelObjectFunctorial
 ‣ ExactCokernelObjectFunctorial( iota_1, nu, iota_2 ) ( operation )

Returns: a morphism C_1 \to C_2

The arguments are an inflation \iota_1:A_1 \to B_1, a morphism \nu: B_1 \to B_2 and an inflation \iota_2:A_2 \to B_2 such that there \mu: A_1 \to A_2 with \comp{\iota_1}{\nu} \sim \comp{\mu}{\iota_2}. The operation delegates to the operation \mathrm{ExactCokernelObjectFunctorialWithGivenExactCokernelObjects}(C_1,\iota_1,\nu,\iota_2,C_2) such that C_1=\mathrm{ExactCokernelObject}(\iota_1), C_2=\mathrm{ExactCokernelObject}(\iota_2).

##### 1.2-10 ExactCokernelObjectFunctorialWithGivenExactCokernelObjects
 ‣ ExactCokernelObjectFunctorialWithGivenExactCokernelObjects( C_1, iota_1, nu, iota_2, C_2 ) ( operation )

Returns: a morphism C_1 \to C_2

The arguments are an object C_1, an inflation \iota_1:A_1 \to B_1, a morphism \nu: B_1 \to B_2, an inflation \iota_2:A_2 \to B_2 and an object C_2 such that C_1=\mathrm{ExactCokernelObject}(\iota_1), C_2=\mathrm{ExactCokernelObject}(\iota_2) and there exists a morphism \mu: A_1 \to A_2 with \comp{\iota_1}{\nu} \sim \comp{\mu}{\iota_2}. The output is the universal morphism \lambda: C_1 \rightarrow C_2 given by the universal property of the cokernel object, i.e., \comp{\pi(\iota_1)}{\lambda} \sim \comp{\nu}{\pi(\iota_2)} where \pi(\iota_1) = \mathrm{ExactCokernelProjection}(\iota_1) and \pi(\iota_2) = \mathrm{ExactCokernelProjection}(\iota_2).

##### 1.2-11 ColiftAlongDeflation
 ‣ ColiftAlongDeflation( pi, tau ) ( operation )

Returns: a morphism C \to T

The arguments are a deflation \pi: B \rightarrow C and a morphism \tau: B \to T such that \tau is coliftable along \pi. That is, \comp{\iota(\pi)}{\tau} \sim 0. The output is the unique colift morphism \lambda:C\to T of \tau along \pi.

##### 1.2-12 ExactKernelObject
 ‣ ExactKernelObject( pi ) ( attribute )

Returns: an object K

The argument is a deflation \pi:B\to C. The output is the kernel object K of \pi.

##### 1.2-13 ExactKernelEmbedding
 ‣ ExactKernelEmbedding( pi ) ( attribute )

Returns: an inflation K\to B

The argument is a deflation \pi:B\to C. The output is an inflation \iota(\pi):K\to B with K=\mathrm{ExactKernelObject}(\pi) such that the pair (\iota(\pi),\pi) defines a conflation.

##### 1.2-14 ExactKernelEmbeddingWithGivenExactKernelObject
 ‣ ExactKernelEmbeddingWithGivenExactKernelObject( pi, K ) ( operation )

Returns: an inflation K\to B

The argument is a deflation \pi:B\to C and an object K=\mathrm{ExactKernelObject}(\pi). The output is an inflation \iota(\pi):K\to B such that the pair (\iota(\pi),\pi) defines a conflation.

##### 1.2-15 ExactKernelLift
 ‣ ExactKernelLift( pi, tau ) ( operation )

Returns: a morphism T \to K

The arguments are a deflation \pi: B \rightarrow C and a test morphism \tau: T \rightarrow B satisfying \comp{\tau}{\pi} \sim 0. The output is the morphism \lambda: T \rightarrow K with K=\mathrm{ExactKernelObject}(\pi) and \lambda is given by the universal property of the kernel object, i.e., \comp{\lambda}{\iota(\pi)} \sim \tau where \iota(\pi) = \mathrm{ExactKernelEmbedding}(\pi).

##### 1.2-16 ExactKernelObjectFunctorial
 ‣ ExactKernelObjectFunctorial( pi_1, mu, pi_2 ) ( operation )

Returns: a morphism K_1 \to K_2

The arguments are a deflation \pi_1:B_1 \to C_1, a morphism \mu: B_1 \to B_2 and a deflation \pi_2: B_2 \to C_2 such that there exists a morphism \nu: C_1 \to C_2 with \comp{\pi_1}{\nu} \sim \comp{\mu}{\pi_2}. The operation delegates to the operation \mathrm{ExactKernelObjectFunctorialWithGivenExactKernelObjects}(K_1,\pi_1,\mu,\pi_2,K_2) where K_1=\mathrm{ExactKernelObject}(\pi_1) and K_2=\mathrm{ExactKernelObject}(\pi_2).

##### 1.2-17 ExactKernelObjectFunctorialWithGivenExactKernelObjects
 ‣ ExactKernelObjectFunctorialWithGivenExactKernelObjects( K_1, pi_1, mu, pi_2, K_2 ) ( operation )

Returns: a morphism K_1 \to K_2

The arguments are an object K_1, a deflation \pi_1:B_1 \to C_1, a morphism \mu: B_1 \to B_2, a deflation \pi_2: B_2 \to C_2 and an object K_2 such that K_1 = \mathrm{ExactKernelObject}(\pi_1), K_2 = \mathrm{ExactKernelObject}(\pi_2) and there exists a morphism \nu: C_1 \to C_2 with \comp{\pi_1}{\nu} \sim \comp{\mu}{\pi_2}. The output is the universal morphism \lambda: K_1 \rightarrow K_2 given by the universal property of the kernel object, i.e., \comp{\lambda}{\iota(\pi_2)} \sim \comp{\iota(\pi_1)}{\mu} where \iota(\pi_1) = \mathrm{ExactKernelEmbedding}(\pi_1) and \iota(\pi_2) = \mathrm{ExactKernelEmbedding}(\pi_2).

##### 1.2-18 LiftAlongInflation
 ‣ LiftAlongInflation( iota, tau ) ( operation )

Returns: a morphism C \to T

The arguments are an inflation \iota: A \rightarrow B and a morphism \tau: T \to B such that \tau is liftable along \iota. That is, \comp{\tau}{\pi(\iota)} \sim 0. The output is the unique lift morphism \lambda:T\to A of \tau along \iota.

##### 1.2-19 Exact Fiber Product

Given a deflation \pi:A\to C and a morphism \alpha:B\to C, an exact fiber product diagram of (\pi,\alpha) is defined by an object A\times_C B, a morphism p_A:A\times_C B\to A and a deflation p_B:A\times_C B\to B such that \comp{p_A}{\pi}\sim \comp{p_B}{\alpha} and for any two morphisms p'_A:T\to A,p'_B:T\to B with \comp{p'_A}{\pi}\sim \comp{p'_B}{\alpha}, there exists a unique morphism u:T\to A\times_C B with \comp{u}{p_A} \sim p'_A and \comp{u}{p_B} \sim p'_B.

##### 1.2-20 ExactFiberProduct
 ‣ ExactFiberProduct( pi, alpha ) ( operation )

Returns: an object

The arguments are a deflation \pi:A\to C and a morphism \alpha:B\to C. The output is the fiber product object A\times_C B of \pi and \alpha.

##### 1.2-21 ProjectionInFirstFactorOfExactFiberProduct
 ‣ ProjectionInFirstFactorOfExactFiberProduct( pi, alpha ) ( operation )

Returns: a morphism p_A:A\times_C B\to A

The arguments are a deflation \pi:A\to C and a morphism \alpha:B\to C. The output is a morphism p_A:A\times_C B \to A which is a part of a fiber product diagram of \pi and \alpha.

##### 1.2-22 ProjectionInSecondFactorOfExactFiberProduct
 ‣ ProjectionInSecondFactorOfExactFiberProduct( pi, alpha ) ( operation )

Returns: a morphism p_B:A\times_C B\to B

The arguments are a deflation \pi:A\to C and a morphism \alpha:B\to C. The output is a morphism p_B:A\times_C B \to B which is a part of a fiber product diagram of \pi and \alpha.

##### 1.2-23 UniversalMorphismIntoExactFiberProduct
 ‣ UniversalMorphismIntoExactFiberProduct( pi, alpha, pprime_A, pprime_B ) ( operation )

Returns: a morphism u:T \to A \times_C B

The arguments are a deflation \pi:A\to C and three morphisms \alpha:B\to C, p'_A:T\to A and p'_B:T\to B such that \comp{p'_A}{\pi} \sim \comp{p'_B}{\alpha}. The output is the universal morphism u:T\to A\times_C B with \comp{u}{p_A} \sim p'_A and \comp{u}{p_B}\sim p'_B.

##### 1.2-24 Exact Pushout

Given an inflation \iota:C\to A and a morphism \alpha:C\to B, an exact pushout diagram of (\iota,\alpha) is defined by an object A\oplus_C B, a morphism q_A:A \to A\oplus_C B and an inflation q_B:B\to A\oplus_C B such that \comp{\iota}{q_A} \sim \comp{\alpha}{q_B} and for any two morphisms q'_A:A\to T,q'_B:B\to T with \comp{\iota}{q'_A}\sim \comp{\alpha}{q'_B}, there exists a unique morphism u: A\oplus_C B \to T with \comp{q_A}{ u}\sim q'_A and \comp{q_B}{ u} \sim q'_B.

##### 1.2-25 ExactPushout
 ‣ ExactPushout( iota, alpha ) ( operation )

Returns: an object

The arguments are an inflation \iota:C\to A and a morphism \alpha:C\to B. The output is the pushout object A\oplus_C B of \iota and \alpha.

##### 1.2-26 InjectionOfFirstCofactorOfExactPushout
 ‣ InjectionOfFirstCofactorOfExactPushout( iota, alpha ) ( operation )

Returns: a morphism A \to A\oplus_C B

The arguments are an inflation \iota:C\to A and a morphism \alpha:C\to B. The output is a morphism q_A:A \to A\oplus_C B which is a part of a pushout diagram of \iota and \alpha.

##### 1.2-27 InjectionOfSecondCofactorOfExactPushout
 ‣ InjectionOfSecondCofactorOfExactPushout( iota, alpha ) ( operation )

Returns: a inflation B \to A\oplus_C B

The arguments are an inflation \iota:C\to A and a morphism \alpha:C\to B. The output is an inflation q_B:B \to A\oplus_C B which is a part of a pushout diagram of \iota and \alpha.

##### 1.2-28 UniversalMorphismFromExactPushout
 ‣ UniversalMorphismFromExactPushout( iota, alpha, qprime_A, qprime_B ) ( operation )

Returns: a morphism A\oplus_C B \to T

The arguments are a inflation \iota:C\to A and three morphisms \alpha:C\to B, q'_A:A\to T and q'_B:B\to T such that \comp{\iota}{ q'_A}\sim \comp{\alpha}{ q'_B}. The output is the universal morphism u:A\oplus_C B\to T with \comp{q_A}{ u}\sim q'_A and \comp{q_B}{u} \sim q'_B.

##### 1.2-29 UniversalMorphismFromExactPushoutWithGivenExactPushout
 ‣ UniversalMorphismFromExactPushoutWithGivenExactPushout( iota, alpha, qprime_A, qprime_B, P ) ( operation )

Returns: a morphism P \to T

The arguments are a inflation \iota:C\to A, three morphisms \alpha:C\to B, q'_A:A\to T, q'_B:B\to T and an object P=A\oplus_C B=\mathrm{ExactPushout}(\iota,\alpha) such that \comp{\iota}{ q'_A}\sim \comp{\alpha}{ q'_B}. The output is the universal morphism u:P \to T with \comp{q_A}{ u}\sim q'_A and \comp{q_B}{u} \sim q'_B.

##### 1.2-30 Exact Categories With Enough E-projectives

Let (\CC,\EE) be an exact category. An object P is called \mathcal{E}-projective if for every morphism \tau:P\to C and every deflation \pi:B\to C, there exists a lift morphism \lambda:P\to B of \tau along \pi, i.e., \comp{\lambda}{\pi}=\tau. The exact category (\CC,\EE) is said to have enough \EE-projectives if for each object A in \CC, there exists a deflation p_A:P_A \to A where P_A is an \EE-projecitve object.

##### 1.2-31 IsExactCategoryWithEnoughExactProjectives
 ‣ IsExactCategoryWithEnoughExactProjectives( C ) ( property )

Returns: true or false

The input is a CAP category. The output is true if \CC is an exact category with respect to some class \EE of short exact sequences and (\CC,\EE) has enough \EE-projectives.

##### 1.2-32 IsExactProjectiveObject
 ‣ IsExactProjectiveObject( P ) ( property )

Returns: a boolian

The argument is an object P in \CC. The output is whether or not P is an \EE-projective object.

##### 1.2-33 ExactProjectiveLift
 ‣ ExactProjectiveLift( tau, pi ) ( operation )

Returns: a morphism \lambda:P\to B

The arguments are a morphism \tau:P\to C where P is an \EE-projective object and a deflation \pi:B\to C. The output is a lift morphism \lambda:P\to B of \tau along \pi, i.e., \comp{\lambda}{\pi} \sim \tau.

##### 1.2-34 SomeExactProjectiveObject
 ‣ SomeExactProjectiveObject( A ) ( attribute )

Returns: an \EE-projective object

The argument is an object A in \CC. The output is an \EE-projective object P_A such that there exists a deflation P_A \to A.

##### 1.2-35 DeflationFromSomeExactProjectiveObject
 ‣ DeflationFromSomeExactProjectiveObject( A ) ( attribute )

Returns: a deflation P_A \to A

The argument is an object A in \CC. The output is a deflation morphism p_A:P_A \to A where P_A = \mathrm{SomeExactProjectiveObject}(A).

##### 1.2-36 Exact Categories With Enough E-injecitves

Let (\CC,\EE) be an exact category. An object I is called \mathcal{E}-injective if for every inflation \iota:A\to B and every morphism \tau:A \to I, there exists a colift morphism of \tau along \iota. The exact category (\CC,\EE) is said to have enough \EE-injectives if for each object A in \CC, there exists an inflation \iota_A:A \to I_A where I_A is an \EE-injective object.

##### 1.2-37 IsExactCategoryWithEnoughExactInjectives
 ‣ IsExactCategoryWithEnoughExactInjectives( C ) ( property )

Returns: true or false

The input is a CAP category. The output is true if \CC is an exact category with respect to some class \EE of short exact sequences and (\CC,\EE) has enough \EE-injectives.

##### 1.2-38 IsExactInjectiveObject
 ‣ IsExactInjectiveObject( I ) ( property )

Returns: a boolian

The argument is an object I in \CC. The output is whether or not I is an \EE-injective object.

##### 1.2-39 ExactInjectiveColift
 ‣ ExactInjectiveColift( iota, tau ) ( operation )

Returns: a morphism \lambda:B \to I

The arguments are an inflation \iota:A\to B and a morphism \tau:A\to I where I is an \EE-injective object. The output is a colift morphism \lambda:B \to I of \tau along \iota, i.e., \comp{\iota}{\lambda} \sim \tau.

##### 1.2-40 SomeExactInjectiveObject
 ‣ SomeExactInjectiveObject( A ) ( operation )

Returns: an \EE-injective object

The argument is an object A in \CC. The output is an \EE-injective object I_A such that there exists an inflation A \to I_A.

##### 1.2-41 InflationIntoSomeExactInjectiveObject
 ‣ InflationIntoSomeExactInjectiveObject( A ) ( operation )

Returns: an inflation A \to I_A

The argument is an object A in \CC. The output is an inflation \iota_A:A \to I_A where I_A = \mathrm{SomeExactInjectiveObject}(A).

##### 1.2-42 IsLiftableAlongDeflationFromSomeExactProjectiveObject
 ‣ IsLiftableAlongDeflationFromSomeExactProjectiveObject( alpha ) ( property )

Returns: true or false

The argument if a morphism \alpha:A\to B in an exact category (\CC,\EE) with enough \EE-projectives. The output is whether or not \alpha lifts along p_B:P_B\to B where p_B=\mathrm{DeflationFromSomeExactProjectiveObject}(B).

##### 1.2-43 LiftAlongDeflationFromSomeExactProjectiveObject
 ‣ LiftAlongDeflationFromSomeExactProjectiveObject( alpha ) ( attribute )

The argument is a morphism \alpha:A\to B such that \alpha lifts along p_B:P_B \to B where p_B=\mathrm{DeflationFromSomeExactProjectiveObject}(B). The output is a lift morphism \lambda: A \to P_B of \alpha along p_B.

##### 1.2-44 IsColiftableAlongInflationIntoSomeExactInjectiveObject
 ‣ IsColiftableAlongInflationIntoSomeExactInjectiveObject( alpha ) ( property )

Returns: true or false

The argument if a morphism \alpha:A\to B in an exact category (\CC,\EE) with enough \EE-injectives. The output is whether or not \alpha colifts along \iota_A:A\to I_A where \iota_A=\mathrm{InflationIntoSomeExactInjectiveObject}(A).

##### 1.2-45 ColiftAlongInflationIntoSomeExactInjectiveObject
 ‣ ColiftAlongInflationIntoSomeExactInjectiveObject( alpha ) ( attribute )

The argument is a morphism \alpha:A\to B such that \alpha colifts along \iota_A:A \to I_A where \iota_A=\mathrm{InflationIntoSomeExactInjectiveObject}(A). The output is a colift morphism \lambda: I_A \to B of \alpha along \iota_A.

##### 1.2-46 IsFrobeniusCategory
 ‣ IsFrobeniusCategory( C ) ( property )

Returns: true or false

The argument is a CAP category. The output is true if

• \CC is exact with respect to some class \EE of short exact sequences,

• (\CC,\EE) has enough \EE-projectives and \EE-injectives,

• an object in \CC is \EE-projective if and only if it is \EE-injective.

##### 1.2-47 SchanuelsIsomorphismByInflationsIntoSomeExactInjectiveObjects
 ‣ SchanuelsIsomorphismByInflationsIntoSomeExactInjectiveObjects( i, s, j, t ) ( operation )

The arguments are an inflation i:A\to I, a deflation s:I \to B, an inflation j:A \to J, a deflation t: J \to C in an exact category (\CC,\EE) such that I and J are \EE-injective objects and the pairs (i,s) and (j,t) are conflations. The output is a morphism u: B \to C such that u becomes an isomorphism in the stable category of \CC by the class of exact injective objects.

##### 1.2-48 SchanuelsIsomorphismByDeflationsFromSomeExactProjectiveObjects
 ‣ SchanuelsIsomorphismByDeflationsFromSomeExactProjectiveObjects( i, s, j, t ) ( operation )

The arguments are an inflation i:A\to I, a deflation s:I \to C, an inflation j:B \to J, a deflation t: J \to C in an exact category (\CC,\EE) such that I and J are \EE-projective objects and the pairs (i,s) and (j,t) are conflations. The output is a morphism u: A \to B such that u becomes an isomorphism in the stable category of \CC by the class of exact projective objects.

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