Any GAP object satisfying IsCapCategoryMorphism can be added to a category and then becomes a morphism in this category. Any morphism can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and to which category it belongs. It knows categorical properties and attributes, and the functions for existential quantifiers can be applied to the morphism.
‣ CapCategory( alpha ) | ( attribute ) |
Returns: a category
The argument is a morphism \(\alpha\). The output is the category \(\mathbf{C}\) to which \(\alpha\) was added.
‣ Source( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its source \(a\).
‣ Range( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its range \(b\).
‣ AddIsMonomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsMonomorphism. \(F: \alpha \mapsto \mathtt{IsMonomorphism}(\alpha)\).
‣ AddIsEpimorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEpimorphism. \(F: \alpha \mapsto \mathtt{IsEpimorphism}(\alpha)\).
‣ AddIsIsomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsIsomorphism. \(F: \alpha \mapsto \mathtt{IsIsomorphism}(\alpha)\).
‣ AddIsSplitMonomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsSplitMonomorphism. \(F: \alpha \mapsto \mathtt{IsSplitMonomorphism}(\alpha)\).
‣ AddIsSplitEpimorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsSplitEpimorphism. \(F: \alpha \mapsto \mathtt{IsSplitEpimorphism}(\alpha)\).
‣ AddIsOne( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsOne. \(F: \alpha \mapsto \mathtt{IsOne}(\alpha)\).
‣ AddIsIdempotent( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsIdempotent. \(F: \alpha \mapsto \mathtt{IsIdempotent}(\alpha)\).
Non-categorical properties are not stable under equivalences of categories.
‣ IsIdenticalToIdentityMorphism( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true if \(\alpha = \mathrm{id}_a\), otherwise the output is false.
‣ AddIsIdenticalToIdentityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsIdenticalToIdentityMorphism. \(F: \alpha \mapsto \mathtt{IsIdenticalToIdentityMorphism}(\alpha)\).
‣ IsIdenticalToZeroMorphism( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true if \(\alpha = 0\), otherwise the output is false.
‣ AddIsIdenticalToZeroMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsIdenticalToZeroMorphism. \(F: \alpha \mapsto \mathtt{IsIdenticalToZeroMorphism }(\alpha)\).
‣ AddIsEndomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEndomorphism. \(F: \alpha \mapsto \mathtt{IsEndomorphism}(\alpha)\).
‣ AddIsAutomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsAutomorphism. \(F: \alpha \mapsto \mathtt{IsAutomorphism}(\alpha)\).
‣ IsCongruentForMorphisms( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is true if \(\alpha \sim_{a,b} \beta\), otherwise the output is false.
‣ AddIsCongruentForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsCongruentForMorphisms. \(F: (\alpha, \beta) \mapsto \mathtt{IsCongruentForMorphisms}(\alpha, \beta)\).
‣ IsEqualForMorphisms( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is true if \(\alpha = \beta\), otherwise the output is false.
‣ AddIsEqualForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEqualForMorphisms. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphisms}(\alpha, \beta)\).
‣ IsEqualForMorphismsOnMor( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha: a \rightarrow b, \beta: c \rightarrow d\). The output is true if \(\alpha = \beta\), otherwise the output is false.
‣ AddIsEqualForMorphismsOnMor( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEqualForMorphismsOnMor. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphismsOnMor}(\alpha, \beta)\).
‣ IsZeroForMorphisms( alpha ) | ( operation ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true if \(\alpha \sim_{a,b} 0\), otherwise the output is false.
‣ AddIsZeroForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsZeroForMorphisms. \(F: \alpha \mapsto \mathtt{IsZeroForMorphisms}(\alpha)\).
‣ AdditionForMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is the addition \(\alpha + \beta\).
‣ AddAdditionForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation AdditionForMorphisms. \(F: (\alpha, \beta) \mapsto \alpha + \beta\).
‣ SubtractionForMorphisms( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is the addition \(\alpha - \beta\).
‣ AddSubtractionForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation SubtractionForMorphisms. \(F: (\alpha, \beta) \mapsto \alpha - \beta\).
‣ AdditiveInverseForMorphisms( alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its additive inverse \(-\alpha\).
‣ AddAdditiveInverseForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation AdditiveInverseForMorphisms. \(F: \alpha \mapsto -\alpha\).
‣ ZeroMorphism( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two objects \(a\) and \(b\). The output is the zero morphism \(0: a \rightarrow b\).
‣ AddZeroMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation ZeroMorphism. \(F: (a,b) \mapsto (0: a \rightarrow b)\).
Subobjects of an object \(c\) are monomorphisms with range \(c\) and a special function for comparision. Similarly, factorobjects of an object \(c\) are epimorphisms with source \(c\) and a special function for comparision.
‣ IsEqualAsSubobjects( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two subobjects \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is true if there exists an isomorphism \(\iota: a \rightarrow b\) such that \(\beta \circ \iota \sim_{a,c} \alpha\), otherwise the output is false.
‣ AddIsEqualAsSubobjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEqualAsSubobjects. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsSubobjects}(\alpha,\beta)\).
‣ IsEqualAsFactorobjects( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two factorobjects \(\alpha: c \rightarrow a\), \(\beta: c \rightarrow b\). The output is true if there exists an isomorphism \(\iota: b \rightarrow a\) such that \(\iota \circ \beta \sim_{c,a} \alpha\), otherwise the output is false.
‣ AddIsEqualAsFactorobjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsEqualAsFactorobjects. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsFactorobjects}(\alpha,\beta)\).
‣ IsDominating( alpha, beta ) | ( operation ) |
Returns: a boolean
In short: Returns true iff \(\alpha\) is smaller than \(\beta\). \(\\ \) Full description: The arguments are two subobjects \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is true if there exists a morphism \(\iota: a \rightarrow b\) such that \(\beta \circ \iota \sim_{a,c} \alpha\), otherwise the output is false.
‣ AddIsDominating( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsDominating. \(F: (\alpha, \beta) \mapsto \mathtt{IsDominating}(\alpha,\beta)\).
‣ IsCodominating( alpha, beta ) | ( operation ) |
Returns: a boolean
In short: Returns true iff \(\alpha\) is smaller than \(\beta\). \(\\ \) Full description: The arguments are two factorobjects \(\alpha: c \rightarrow a\), \(\beta: c \rightarrow b\). The output is true if there exists a morphism \(\iota: b \rightarrow a\) such that \(\iota \circ \beta \sim_{c,a} \alpha\), otherwise the output is false.
‣ AddIsCodominating( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsCodominating. \(F: (\alpha, \beta) \mapsto \mathtt{IsCodominating}(\alpha,\beta)\).
‣ IdentityMorphism( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a,a)\)
The argument is an object \(a\). The output is its identity morphism \(\mathrm{id}_a\).
‣ AddIdentityMorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IdentityMorphism. \(F: a \mapsto \mathrm{id}_a\).
‣ PreCompose( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, c )\)
The arguments are two morphisms \(\alpha: a \rightarrow b\), \(\beta: b \rightarrow c\). The output is the composition \(\beta \circ \alpha: a \rightarrow c\).
‣ PreCompose( L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a_1, a_{n+1})\)
This is a convenience method. The argument is a list of morphisms \(L = ( \alpha_1: a_1 \rightarrow a_2, \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \rightarrow a_{n+1} )\). The output is the composition \(\alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) )\).
‣ AddPreCompose( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation PreCompose. \(F: (\alpha, \beta) \mapsto \beta \circ \alpha\).
‣ PostCompose( beta, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, c )\)
The arguments are two morphisms \(\beta: b \rightarrow c\), \(\alpha: a \rightarrow b\). The output is the composition \(\beta \circ \alpha: a \rightarrow c\).
‣ PostCompose( L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a_1, a_{n+1})\)
This is a convenience method. The argument is a list of morphisms \(L = ( \alpha_n: a_n \rightarrow a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n, \dots, \alpha_1: a_1 \rightarrow a_2 )\). The output is the composition \(((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1\).
‣ AddPostCompose( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation PostCompose. \(F: (\alpha, \beta) \mapsto \alpha \circ \beta\).
‣ IsWellDefinedForMorphisms( alpha ) | ( operation ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true if \(\alpha\) is well-defined, otherwise the output is false.
‣ AddIsWellDefinedForMorphisms( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation IsWellDefinedForMorphisms. \(F: \alpha \mapsto \mathtt{IsWellDefinedForMorphisms}( \alpha )\).
‣ LiftAlongMonomorphism( iota, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(t,k)\)
The arguments are a monomorphism \(\iota: k \hookrightarrow a\) and a morphism \(\tau: t \rightarrow a\) such that there is a morphism \(u: t \rightarrow k\) with \(\iota \circ u \sim_{t,a} \tau\). The output is such a \(u\).
‣ AddLiftAlongMonomorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation LiftAlongMonomorphism. The function \(F\) maps a pair \((\iota, \tau)\) to a lift \(u\) if it exists, and to fail otherwise.
‣ ColiftAlongEpimorphism( epsilon, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(c,t)\)
The arguments are an epimorphism \(\epsilon: a \rightarrow c\) and a morphism \(\tau: a \rightarrow t\) such that there is a morphism \(u: c \rightarrow t\) with \(u \circ \epsilon \sim_{a,t} \tau\). The output is such a \(u\).
‣ AddColiftAlongEpimorphism( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation ColiftAlongEpimorphism. The function \(F\) maps a pair \((\epsilon, \tau)\) to a lift \(u\) if it exists, and to fail otherwise.
For any pair of morphisms \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\), we call each morphism \(\alpha / \beta: a \rightarrow b\) such that \(\beta \circ (\alpha / \beta) \sim_{a,c} \alpha\) a lift of \(\alpha\) along \(\beta\).
For any pair of morphisms \(\alpha: a \rightarrow c\), \(\beta: a \rightarrow b\), we call each morphism \(\alpha \backslash \beta: c \rightarrow b\) such that \((\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta\) a colift of \(\beta\) along \(\alpha\).
Note that such lifts (or colifts) do not have to be unique. So in general, we do not expect that algorithms computing lifts (or colifts) do this in a functorial way. Thus the operations \(\mathtt{Lift}\) and \(\mathtt{Colift}\) are not regarded as categorical operations, but only as set-theoretic operations.
‣ Lift( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b) + \{ \mathtt{fail} \}\)
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\) such that there is a lift \(\alpha / \beta: a \rightarrow b\) of \(\alpha\) along \(\beta\), i.e., a morphism such that \(\beta \circ (\alpha / \beta) \sim_{a,c} \alpha\). The output is such a lift or \(\mathtt{fail}\) if it doesn't exist.
‣ AddLift( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation Lift. The function \(F\) maps a pair \((\alpha, \beta)\) to a lift \(\alpha / \beta\) if it exists, and to fail otherwise.
‣ Colift( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(c,b) + \{ \mathtt{fail} \}\)
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: a \rightarrow b\) such that there is a colift \(\alpha \backslash \beta: c \rightarrow b\) of \(\beta\) along \(\alpha\)., i.e., a morphism such that \((\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta\). The output is such a colift or \(\mathtt{fail}\) if it doesn't exist.
‣ AddColift( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation Colift. The function \(F\) maps a pair \((\alpha, \beta)\) to a colift \(\alpha \backslash \beta\) if it exists, and to fail otherwise.
Let \(\alpha: a \rightarrow b\) be a morphism. An inverse of \(\alpha\) is a morphism \(\alpha^{-1}: b \rightarrow a\) such that \(\alpha \circ \alpha^{-1} \sim_{b,b} \mathrm{id}_b\) and \(\alpha^{-1} \circ \alpha \sim_{a,a} \mathrm{id}_a\).
‣ AddInverse( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation Inverse. \(F: \alpha \mapsto \alpha^{-1}\).
‣ IsEqualForCacheForMorphisms( phi, psi ) | ( operation ) |
Returns: true or false
Compares two objects in the cache
‣ AddIsEqualForCacheForMorphisms( c, F ) | ( operation ) |
Returns: northing
By default, CAP uses caches to store the values of Categorical operations. To get a value out of the cache, one needs to compare the input of a basic operation with its previous input. To compare morphisms in the category, IsEqualForCacheForMorphism is used. By default this is an alias for IsEqualForMorphismsOnMor, where fail is substituted by false. If you add a function, this function used instead. A function \(F: a,b \mapsto bool\) is expected here. The output has to be true or false. Fail is not allowed in this context.
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