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Quelle CategoryMorphismsOperations.gd Sprache: unbekannt |
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# SPDX-License-Identifier: GPL-2.0-or-later
# CAP: Categories, Algorithms, Programming # # Declarations # #! @Chapter Morphisms ################################### ## #! @Section Morphism constructors ## ################################### #! @Description #! The arguments are two objects $S$ and $T$ in a category, #! and a morphism datum $a$ (type and semantics of the morphism datum depend on the category). #! The output is a morphism in $\mathrm{Hom}(S,T)$ defined by $a$. #! Note that by default this CAP operation is not cached. You can change this behaviour #! by calling `SetCachingToWeak( C, "MorphismConstructor" )` resp. `SetCachingToCrisp( C, #! @Returns a morphism in $\mathrm{Hom}(S,T)$ #! @Arguments S, a, T DeclareOperation( "MorphismConstructor", [ IsCapCategoryObject, IsObject, IsCapCategoryObject ] ); #! @Description #! The argument is a CAP category morphism <A>mor</A>. #! The output is a datum which can be used to construct <A>mor</A>, that is, #! `IsEqualForMorphisms( `<A>mor</A>`, MorphismConstructor( Source( `<A>mor</A>` ), MorphismDat #! Note that by default this CAP operation is not cached. You can change this behaviour #! by calling `SetCachingToWeak( C, "MorphismDatum" )` resp. `SetCachingToCrisp( C, "Morph #! @Returns depends on the category #! @Arguments mor DeclareAttribute( "MorphismDatum", IsCapCategoryMorphism ); ################################### ## #! @Section Categorical Properties of Morphisms ## ################################### #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is a monomorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsMonomorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is an epimorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsEpimorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is an isomorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsIsomorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is a split monomorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsSplitMonomorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is a split epimorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsSplitEpimorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha: a \rightarrow a$. #! The output is <C>true</C> if $\alpha$ is congruent to the identity of $a$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsOne", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha: a \rightarrow a$. #! The output is <C>true</C> if $\alpha^2 \sim_{a,a} \alpha$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsIdempotent", IsCapCategoryMorphism ); ################################### ## #! @Section Random Morphisms ## ################################### #! CAP provides two principal methods to generate random morphisms with or without fixed sourc #! * <E>By integers</E>: The integer is simply a parameter that can be used to create a random morphis #! * <E>By lists</E>: The list is used when creating a random morphism would need more than one parame #! flexibility at the expense of the genericity of the methods. This happens because lists that #! some category may be not valid for other categories. Hence, these operations are not thought #! generic categorical algorithms. #! @Description #! The arguments are an object $a$ in a category $C$ and an integer $n$. #! The output is a random morphism $\alpha: a \rightarrow b$ for some object $b$ in $C$. #! If $C$ is equipped with the methods <C>RandomObjectByInteger</C> and <C>RandomMorphismWithFix #! and $C$ is an Ab-category, then <C>RandomMorphismWithFixedSourceByInteger</C>$(C,a,n)$ ca #! <C>RandomMorphismWithFixedSourceAndRangeByInteger</C>($C$,$a$,$b$,$1$+<C>Log2Int</C>($ #! $b$ is computed via <C>RandomObjectByInteger</C>($C$,$n$). #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments a, n DeclareOperation( "RandomMorphismWithFixedSourceByInteger", [ IsCapCategoryObject, IsInt ] ); #! @Description #! The arguments are an object $a$ in a category $C$ and a list $L$. #! The output is a random morphism $\alpha: a \rightarrow b$ for some object $b$ in $C$. #! If $C$ is equipped with the methods <C>RandomObjectByList</C> and <C>RandomMorphismWithFixedS #! and $C$ is an Ab-category, then <C>RandomMorphismWithFixedSourceByList</C>$(C,a,L)$ can be #! <C>RandomMorphismWithFixedSourceAndRangeByList</C>($C,a,b,L[2]$) where #! $b$ is computed via <C>RandomObjectByList</C>($C,L[1]$). #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments a, L DeclareOperation( "RandomMorphismWithFixedSourceByList", [ IsCapCategoryObject, IsList ] ); #! @Description #! The arguments are an object $b$ in a category $C$ and an integer $n$. #! The output is a random morphism $\alpha: a \rightarrow b$ for some object $a$ in $C$. #! If $C$ is equipped with the methods <C>RandomObjectByInteger</C> and <C>RandomMorphismWithFix #! and $C$ is an Ab-category, then <C>RandomMorphismWithFixedRangeByInteger</C>$(C,b,n)$ can #! <C>RandomMorphismWithFixedSourceAndRangeByInteger</C>($C$,$a$,$b$,$1$+<C>Log2Int</C>($ #! $a$ is computed via <C>RandomObjectByInteger</C>($C$,$n$). #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments b, n DeclareOperation( "RandomMorphismWithFixedRangeByInteger", [ IsCapCategoryObject, IsInt ] ); #! @Description #! The arguments are an object $b$ in a category $C$ and a list $L$. #! The output is a random morphism $\alpha: a \rightarrow b$ for some object $a$ in $C$. #! If $C$ is equipped with the methods <C>RandomObjectByList</C> and <C>RandomMorphismWithFixedS #! and $C$ is an Ab-category, then <C>RandomMorphismWithFixedRangeByList</C>$(C,b,L)$ can be d #! <C>RandomMorphismWithFixedSourceAndRangeByList</C>($C,a,b,L[2]$) where #! $a$ is computed via <C>RandomObjectByList</C>($C,L[1]$). #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments b, L DeclareOperation( "RandomMorphismWithFixedRangeByList", [ IsCapCategoryObject, IsList ] ); #! @Description #! The arguments are two objects $a$ and $b$ in a category $C$ and an integer $n$. #! The output is a random morphism $\alpha: a \rightarrow b$ in $C$. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments a, b, n DeclareOperation( "RandomMorphismWithFixedSourceAndRangeByInteger", [ IsCapCategoryObject, IsCapCategoryObject, IsInt ] ); #! @Description #! This operation is not a CAP basic operation #! The arguments are two objects $a$ and $b$ in a category $C$ and a list $L$. #! The output is a random morphism $\alpha: a \rightarrow b$ in $C$. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments a, b, L DeclareOperation( "RandomMorphismWithFixedSourceAndRangeByList", [ IsCapCategoryObject, IsCapCategoryObject, IsList ] ); #! @Description #! The arguments are a category $C$ and an integer $n$. #! The output is a random morphism in $C$. #! The operation can be derived in three different ways: #! - If $C$ is equipped with the methods <C>RandomObjectByInteger</C> and <C>RandomMorphismWithFi #! and $C$ is an Ab-category, then <C>RandomMorphism</C>$(C,n)$ can be derived as #! <C>RandomMorphismWithFixedSourceAndRangeByInteger</C>($C,a,b$,$1$+<C>Log2Int</C>($n$)) #! $a$ and $b$ are computed via <C>RandomObjectByInteger</C>($C,n$). #! - If $C$ is equipped with the methods <C>RandomObjectByInteger</C> and <C>RandomMorphismWithFi #! then <C>RandomMorphism</C>$(C,n)$ can be derived as #! <C>RandomMorphismWithFixedSourceByInteger</C>($C,a,1$+<C>Log2Int</C>($n$)) where #! $a$ is computed via <C>RandomObjectByInteger</C>($C,n$). #! - If $C$ is equipped with the methods <C>RandomObjectByInteger</C> and <C>RandomMorphismWithFi #! then <C>RandomMorphism</C>$(C,n)$ can be derived as #! <C>RandomMorphismWithFixedRangeByInteger</C>($C,b,1$+<C>Log2Int</C>($n$)) where #! $b$ is computed via <C>RandomObjectByInteger</C>($C,n$). #! @Returns a morphism in $C$ #! @Arguments C, n DeclareOperation( "RandomMorphismByInteger", [ IsCapCategory, IsInt ] ); #! @Description #! The arguments are a category $C$ and a list $L$. #! The output is a random morphism in $C$. #! The operation can be derived in three different ways: #! - If $C$ is equipped with the methods <C>RandomObjectByList</C> and <C>RandomMorphismWithFixed #! and $C$ is an Ab-category, then <C>RandomMorphism</C>$(C,L)$ can be derived as #! <C>RandomMorphismWithFixedSourceAndRangeByList</C>($C,a,b,L[3]$)) where #! $a$ and $b$ are computed via <C>RandomObjectByList</C>($C,L[i]$) for $i=1,2$ respectively. #! - If $C$ is equipped with the methods <C>RandomObjectByList</C> and <C>RandomMorphismWithFixed #! then <C>RandomMorphism</C>$(C,L)$ can be derived as #! <C>RandomMorphismWithFixedSourceByList</C>($C,a,L[2]$) where #! $a$ is computed via <C>RandomObjectByList</C>($C,L[1]$). #! - If $C$ is equipped with the methods <C>RandomObjectByList</C> and <C>RandomMorphismWithFixed #! then <C>RandomMorphism</C>$(C,L)$ can be derived as #! <C>RandomMorphismWithFixedRangeByList</C>($C,b,L[2]$) where #! $b$ is computed via <C>RandomObjectByList</C>($C,L[1]$). #! @Returns a morphism in $C$ #! @Arguments C, L DeclareOperation( "RandomMorphismByList", [ IsCapCategory, IsList ] ); #! @BeginGroup #! @Description #! These are convenient methods and they, depending on the input, delegate to one of the above me # @Returns an object, morphism in $C$ #! @Arguments a, n DeclareOperation( "RandomMorphismWithFixedSource", [ IsCapCategoryObject, IsInt ] ); #! @Arguments a, L DeclareOperation( "RandomMorphismWithFixedSource", [ IsCapCategoryObject, IsList ] ); #! @Arguments b, n DeclareOperation( "RandomMorphismWithFixedRange", [ IsCapCategoryObject, IsInt ] ); #! @Arguments b, L DeclareOperation( "RandomMorphismWithFixedRange", [ IsCapCategoryObject, IsList ] ); #! @Arguments a, b, n DeclareOperation( "RandomMorphismWithFixedSourceAndRange", [ IsCapCategoryObject, Is #! @Arguments a, b, L DeclareOperation( "RandomMorphismWithFixedSourceAndRange", [ IsCapCategoryObject, Is #! @Arguments a, b, n DeclareOperation( "RandomMorphism", [ IsCapCategoryObject, IsCapCategoryObject, IsInt #! @Arguments a, b, L DeclareOperation( "RandomMorphism", [ IsCapCategoryObject, IsCapCategoryObject, IsLis #! @Arguments C, n DeclareOperation( "RandomMorphism", [ IsCapCategory, IsInt ] ); #! @Arguments C, L DeclareOperation( "RandomMorphism", [ IsCapCategory, IsList ] ); #! @EndGroup ################################### ## #! @Section Non-Categorical Properties of Morphisms ## ################################### #! Non-categorical properties are not stable under equivalences of categories. #! @Description #! The argument is a morphism $\alpha: a \rightarrow b$. #! The output is <C>true</C> if $\alpha = \mathrm{id}_a$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsEqualToIdentityMorphism", IsCapCategoryMorphism ); #! @Description #! The argument is a morphism $\alpha: a \rightarrow b$. #! The output is <C>true</C> if $\alpha = 0$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsEqualToZeroMorphism", IsCapCategoryMorphism ); ## This is not a categorical property because non-endomorphisms ## can be mapped to endomorphisms under equivalences of categories. #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is an endomorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsEndomorphism", IsCapCategoryMorphism ); ## This is not a categorical property because non-endomorphisms ## can be mapped to endomorphisms under equivalences of categories. #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is an automorphism, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsAutomorphism", IsCapCategoryMorphism ); ################################### ## #! @Section Equality and Congruence for Morphisms ## ################################### #! @Description #! The arguments are two morphisms $\alpha, \beta: a \rightarrow b$. #! The output is <C>true</C> if $\alpha \sim_{a,b} \beta$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsCongruentForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha, \beta: a \rightarrow b$. #! The output is <C>true</C> if $\alpha = \beta$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsEqualForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow b, \beta: c \rightarrow d$. #! The output is <C>true</C> if $\alpha = \beta$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsEqualForMorphismsOnMor", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); ################################### ## #! @Section Basic Operations for Morphisms in Ab-Categories ## ################################### #! @Description #! The argument is a morphism $\alpha: a \rightarrow b$. #! The output is <C>true</C> if $\alpha \sim_{a,b} 0$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareProperty( "IsZeroForMorphisms", IsCapCategoryMorphism ); DeclareProperty( "IsZero", IsCapCategoryMorphism ); DeclareOperation( "+", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); DeclareOperation( "-", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha, \beta: a \rightarrow b$. #! The output is the addition $\alpha + \beta$. #! Note: The addition has to be compatible with the congruence of morphisms. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments alpha, beta DeclareOperation( "AdditionForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha, \beta: a \rightarrow b$. #! The output is the addition $\alpha - \beta$. #! Note: The addition has to be compatible with the congruence of morphisms. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments alpha, beta DeclareOperation( "SubtractionForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The argument is a morphism $\alpha: a \rightarrow b$. #! The output is its additive inverse $-\alpha$. #! Note: The addition has to be compatible with the congruence of morphisms. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments alpha DeclareAttribute( "AdditiveInverseForMorphisms", IsCapCategoryMorphism ); DeclareAttribute( "AdditiveInverse", IsCapCategoryMorphism ); #! @Description #! The arguments are an element $r$ of a commutative ring #! and a morphism $\alpha: a \rightarrow b$. #! The output is the multiplication with the ring element $r \cdot \alpha$. #! Note: The multiplication has to be compatible with the congruence of morphisms. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments r, alpha DeclareOperation( "MultiplyWithElementOfCommutativeRingForMorphisms", [ IsRingElement, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience method. It has two arguments. #! The first argument is either a rational number $q$ #! or an element $r$ of a commutative ring $R$. #! The second argument is a morphism $\alpha: a \rightarrow b$ in a linear category #! over the commutative ring $R$. #! In the case where the first element is a rational number, this method tries to interpret $q$ as #! <C>R!.interpret_rationals_func</C>. If no such interpretation #! exists, this method throws an error. #! The output is the multiplication with the ring element $r \cdot \alpha$. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments r, alpha DeclareOperation( "*", [ IsRingElement, IsCapCategoryMorphism ] ); DeclareOperation( "*", [ IsCapCategoryMorphism, IsRingElement ] ); ################################### ## ## Zero Morphism ## ################################### #! @Description #! The arguments are two objects $a$ and $b$. #! The output is the zero morphism $0: a \rightarrow b$. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments a, b DeclareOperation( "ZeroMorphism", [ IsCapCategoryObject, IsCapCategoryObject ] ); ################################### ## #! @Section Subobject and Factorobject Operations ## ################################### #! 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. ## TODO # @Description # This is a synonym for <C>IsMonomorphism</C>. DeclareSynonymAttr( "IsSubobject", IsMonomorphism ); ## TODO # @Description # This is a synonym for <C>IsEpimorphism</C>. DeclareSynonymAttr( "IsFactorobject", IsEpimorphism ); #! @Description #! The arguments are two subobjects $\alpha: a \rightarrow c$, $\beta: b \rightarrow c$. #! The output is <C>true</C> if there exists an isomorphism $\iota: a \rightarrow b$ #! such that $\beta \circ \iota \sim_{a,c} \alpha$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsEqualAsSubobjects", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two factorobjects $\alpha: c \rightarrow a$, $\beta: c \rightarrow b$. #! The output is <C>true</C> if there exists an isomorphism $\iota: b \rightarrow a$ #! such that $\iota \circ \beta \sim_{c,a} \alpha$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsEqualAsFactorobjects", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! @BeginLatexOnly #! \begin{center} #! \begin{tikzpicture} #! \def\w{2} #! \def\h{1} #! \node (a) at (0,\h) {$a$}; #! \node (b) at (0,-\h) {$b$}; #! \node (c) at (\w,0) {$c$}; #! \draw[right hook-latex] (a) to node[pos=0.45, above] {$\alpha$} (c); #! \draw[right hook-latex] (b) to node[pos=0.45, below] {$\beta$} (c); #! \draw[-latex, dashed] (a) to node[pos=0.45, left] {$\exists \iota$} (b); #! \end{tikzpicture} #! \end{center} #! @EndLatexOnly #! In short: Returns <C>true</C> iff $\alpha$ is smaller than $\beta$. #! Full description: The arguments are two subobjects $\alpha: a \rightarrow c$, $\beta: b \ri #! The output is <C>true</C> if there exists a morphism $\iota: a \rightarrow b$ #! such that $\beta \circ \iota \sim_{a,c} \alpha$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsDominating", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! @BeginLatexOnly #! \begin{center} #! \begin{tikzpicture} #! \def\w{2} #! \def\h{1} #! \node (c) at (0,0) {$c$}; #! \node (a) at (\w,\h) {$a$}; #! \node (b) at (\w,-\h) {$b$}; #! \draw[-twohead] (c) to node[pos=0.45, above] {$\alpha$} (a); #! \draw[-twohead] (c) to node[pos=0.45, below] {$\beta$} (b); #! \draw[-latex, dashed] (b) to node[pos=0.45, right] {$\exists \iota$} (a); #! \end{tikzpicture} #! \end{center} #! @EndLatexOnly #! In short: Returns <C>true</C> iff $\alpha$ is smaller than $\beta$. #! Full description: #! The arguments are two factorobjects $\alpha: c \rightarrow a$, $\beta: c \rightarrow b$. #! The output is <C>true</C> if there exists a morphism $\iota: b \rightarrow a$ #! such that $\iota \circ \beta \sim_{c,a} \alpha$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsCodominating", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); ################################### ## #! @Section Identity Morphism and Composition of Morphisms ## ################################### #! @Description #! The argument is an object $a$. #! The output is its identity morphism $\mathrm{id}_a$. #! @Returns a morphism in $\mathrm{Hom}(a,a)$ #! @Arguments a DeclareAttribute( "IdentityMorphism", IsCapCategoryObject ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow b$, $\beta: b \rightarrow c$. #! The output is the composition $\beta \circ \alpha: a \rightarrow c$. #! @Returns a morphism in $\mathrm{Hom}( a, c )$ #! @Arguments alpha, beta DeclareOperation( "PreCompose", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! 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 \r #! The output is the composition #! $\alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) )$. #! @Returns a morphism in $\mathrm{Hom}(a_1, a_{n+1})$ #! @Arguments L DeclareOperation( "PreCompose", [ IsList ] ); #! @Description #! The arguments are two objects <A>s</A> = $a_1$, <A>r</A> = $a_{n+1}$, and a list of morphisms #! $L = ( \alpha_1: a_1 \rightarrow a_2, \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \r #! The output is the composition #! $\alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) )$. #! If $L$ is empty, then $s$ must be equal to $r$ and the output is congruent to the identity morphi #! @Returns a morphism in $\mathrm{Hom}(s, r)$ #! @Arguments s, L, r DeclareOperation( "PreComposeList", [ IsCapCategoryObject, IsList, IsCapCategoryObject ] ); #! @Description #! The arguments are two morphisms $\beta: b \rightarrow c$, $\alpha: a \rightarrow b$. #! The output is the composition $\beta \circ \alpha: a \rightarrow c$. #! @Returns a morphism in $\mathrm{Hom}( a, c )$ #! @Arguments beta, alpha DeclareOperation( "PostCompose", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! 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, \a #! The output is the composition #! $((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1$. #! @Returns a morphism in $\mathrm{Hom}(a_1, a_{n+1})$ #! @Arguments L DeclareOperation( "PostCompose", [ IsList ] ); #! @Description #! The arguments are two objects <A>s</A> = $a_1$, <A>r</A> = $a_{n+1}$, and a list of morphisms #! $L = ( \alpha_n: a_n \rightarrow a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n, \dots, \a #! The output is the composition #! $((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1$. #! If $L$ is empty, then $s$ must be equal to $r$ and the output is congruent to the identity morphi #! @Returns a morphism in $\mathrm{Hom}(s, r)$ #! @Arguments s, L, r DeclareOperation( "PostComposeList", [ IsCapCategoryObject, IsList, IsCapCategoryObject ] ); #! @Description #! The arguments are two objects <A>s</A>, <A>r</A> and a list <A>morphisms</A> of morphisms from <A>s</A> to <A>r</ #! The output is the sum of all elements in <A>morphisms</A>, or the zero-morphism from <A>s</A> to <A>r</A> #! if <A>morphisms</A> is empty. #! @Returns a morphism in $\mathrm{Hom}(s,r)$ #! @Arguments s, morphisms, r DeclareOperation( "SumOfMorphisms", [ IsCapCategoryObject, IsList, IsCapCategoryObject ] ); #! @Description #! The arguments are two objects <A>s</A>, <A>r</A> in some linear category over a ring $R$, #! a list <A>coeffs</A> of ring elements in $R$ and a list <A>mors</A> of morphisms from <A>s</A> to <A>r</A>. #! The output is the linear combination of the morphisms in <A>mors</A> with respect to the coeffici #! or the zero morphism from <A>s</A> to <A>r</A> if <A>coeffs</A> and <A>mors</A> are the empty lists. #! @Returns a morphism in $\mathrm{Hom}(s,r)$ #! @Arguments s, coeffs, mors, r DeclareOperation( "LinearCombinationOfMorphisms", [ IsCapCategoryObject, IsList, IsList, IsCapCategoryObject ] ); ################################### ## #! @Section Well-Definedness of Morphisms ## ################################### #! @Description #! The argument is a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is well-defined, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha DeclareOperation( "IsWellDefinedForMorphisms", [ IsCapCategoryMorphism ] ); #! @Description #! The arguments are two well-defined objects $S$ and $T$ and a morphism $\alpha$. #! The output is <C>true</C> if $\alpha$ is a well-defined morphism from $S$ to $T$, #! otherwise the output is <C>false</C>. #! @Returns a boolean #! @Arguments source, alpha, range DeclareOperation( "IsWellDefinedForMorphismsWithGivenSourceAndRange", [ IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryObject ] ); ################################### ## #! @Section Lift/Colift ## ################################### #! * 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 <Emph>lift of $\alpha$ along $\beta$</Emph> #! @BeginLatexOnly #! \begin{center} #! \begin{tikzpicture} #! \def\w{2} #! \def\h{1} #! \node (a) at (0,\h) {$a$}; #! \node (b) at (0,-\h) {$b$}; #! \node (c) at (\w,0) {$c$}; #! \draw[-latex] (a) to node[pos=0.45, above] {$\alpha$} (c); #! \draw[-latex] (b) to node[pos=0.45, below] {$\beta$} (c); #! \draw[-latex, dashed] (a) to node[pos=0.45, left] {$\alpha/\beta$} (b); #! \end{tikzpicture} #! \end{center} #! @EndLatexOnly #! * 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 <Emph> colift of $\beta$ along $\ #! @BeginLatexOnly #! \begin{center} #! \begin{tikzpicture} #! \def\w{2} #! \def\h{1} #! \node (a) at (0,0) {$a$}; #! \node (c) at (\w,\h) {$c$}; #! \node (b) at (\w,-\h) {$b$}; #! \draw[-latex] (a) to node[pos=0.45, above] {$\alpha$} (c); #! \draw[-latex] (a) to node[pos=0.45, below] {$\beta$} (b); #! \draw[-latex, dashed] (c) to node[pos=0.45, right] {$\alpha \backslash \beta$} (b); #! \end{tikzpicture} #! \end{center} #! @EndLatexOnly #! 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. #! @Description #! 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 $u$. #! @Returns a morphism in $\mathrm{Hom}(t,k)$ #! @Arguments iota, tau DeclareOperation( "LiftAlongMonomorphism", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! 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 $u$. #! @Returns a morphism in $\mathrm{Hom}(c,t)$ #! @Arguments epsilon, tau DeclareOperation( "ColiftAlongEpimorphism", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are a monomorphism $\iota: k \hookrightarrow a$ #! and a morphism $\tau: t \rightarrow a$. #! The output is <C>true</C> if there exists #! a morphism $u: t \rightarrow k$ with #! $\iota \circ u \sim_{t,a} \tau$. #! Otherwise, the output is <C>false</C>. #! @Returns a boolean #! @Arguments iota, tau DeclareOperation( "IsLiftableAlongMonomorphism", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are an epimorphism $\epsilon: a \rightarrow c$ #! and a morphism $\tau: a \rightarrow t$. #! The output is <C>true</C> if there exists #! a morphism $u: c \rightarrow t$ with #! $u \circ \epsilon \sim_{a,t} \tau$. #! Otherwise, the output is <C>false</C>. #! @Returns a boolean #! @Arguments epsilon, tau DeclareOperation( "IsColiftableAlongEpimorphism", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: b \rightarrow c$ #! such that a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$ exists. #! The output is such a lift $\alpha / \beta: a \rightarrow b$. #! Recall that a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$ is #! a morphism such that $\beta \circ (\alpha / \beta) \sim_{a,c} \alpha$. #! @Returns a morphism in $\mathrm{Hom}(a,b)$ #! @Arguments alpha, beta DeclareOperation( "Lift", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience operation. #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: b \rightarrow c$. #! The output is a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$ #! if such a lift exists or $\mathtt{fail}$ if it doesn't. #! Recall that a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$ is #! a morphism such that $\beta \circ (\alpha / \beta) \sim_{a,c} \alpha$. #! @Returns a morphism in $\mathrm{Hom}(a,b) + \{ \mathtt{fail} \}$ #! @Arguments alpha, beta DeclareOperation( "LiftOrFail", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: b \rightarrow c$. #! The output is <C>true</C> if there exists #! a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$, i.e., #! a morphism such that $\beta \circ (\alpha / \beta) \sim_{a,c} \alpha$. #! Otherwise, the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsLiftable", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: a \rightarrow b$ #! such that a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$ exis #! The output is such a colift $\alpha \backslash \beta: c \rightarrow b$. #! Recall that a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$ is #! a morphism such that $(\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta$. #! @Returns a morphism in $\mathrm{Hom}(c,b)$ #! @Arguments alpha, beta DeclareOperation( "Colift", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience operation. #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: a \rightarrow b$. #! The output is a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$ #! if such a colift exists or $\mathtt{fail}$ if it doesn't. #! Recall that a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$ is #! a morphism such that $(\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta$. #! @Returns a morphism in $\mathrm{Hom}(c,b) + \{ \mathtt{fail} \}$ #! @Arguments alpha, beta DeclareOperation( "ColiftOrFail", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow c$, $\beta: a \rightarrow b$. #! The output is <C>true</C> if there exists #! a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$., i.e., #! a morphism such that $(\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta$. #! Otherwise, the output is <C>false</C>. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "IsColiftable", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #################################### ## #! @Section Inverses ## #################################### #! 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 $\alpha^{-1} \circ \alpha \sim_{a,a} \mathrm{id}_a$. #! @BeginLatexOnly #! \begin{center} #! \begin{tikzpicture} #! \def\w{2} #! \def\h{1} #! \node (a) at (0,0) {$a$}; #! \node (b) at (\w,0) {$b$}; #! \draw[-latex] (a) to node[pos=0.45, above] {$\alpha$} (b); #! \draw[-latex] (b) to [out = -135, in = -45] node[pos=0.45, below] {$\alpha^{-1}$} (a); #! \draw [-latex] (a.135) arc (45:45+280:4mm) node[pos=0.5,left] {$\mathrm{id}_a$} (a); #! \draw [-latex] (b.45) arc (-240:-240-280:4mm) node[pos=0.5,right] {$\mathrm{id}_b$} ( #! \end{tikzpicture} #! \end{center} #! @EndLatexOnly #! @Description #! The argument is an isomorphism $\alpha: a \rightarrow b$. #! The output is its inverse $\alpha^{-1}: b \rightarrow a$. #! @Returns a morphism in $\mathrm{Hom}(b,a)$ #! @Arguments alpha DeclareOperation( "InverseForMorphisms", [ IsCapCategoryMorphism ] ); #! @Description #! The argument is a split-epimorphism $\alpha: a \rightarrow b$. #! The output is a pre-inverse $\iota: b \rightarrow a$ of $\alpha$, #! i.e., $\iota$ satisfies $\alpha \circ \iota \sim_{b,b} \mathrm{id}_b$. #! The morphism $\iota$ is also known as a section or a right-inverse of $\alpha$. #! @Returns a morphism in $\mathrm{Hom}(b,a)$ #! @Arguments alpha DeclareOperation( "PreInverseForMorphisms", [ IsCapCategoryMorphism ] ); #! @Description #! The argument is a split-monomorphism $\alpha: a \rightarrow b$. #! The output is a post-inverse $\pi: b \rightarrow a$ of $\alpha$, #! i.e., $\pi$ satisfies $\pi \circ \alpha \sim_{a,a} \mathrm{id}_a$. #! The morphism $\pi$ is also known as a contraction or a left-inverse of $\alpha$. #! @Returns a morphism in $\mathrm{Hom}(b,a)$ #! @Arguments alpha DeclareOperation( "PostInverseForMorphisms", [ IsCapCategoryMorphism ] ); ################################### ## #! @Section Tool functions for caches ## ################################### #! @Description #! 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, IsEqualForCacheForMorphis #! used. By default, IsEqualForCacheForMorphisms falls back to IsEqualForCache (see ToolsF #! which in turn defaults to recursive comparison for lists and `IsIdenticalObj` in all other c #! If you add a function via `AddIsEqualForCacheForMorphisms`, that function is used instead #! A function $F: a,b \mapsto bool$ is expected there. The output has to be #! true or false. Fail is not allowed in this context. #! @Arguments phi, psi #! @Returns true or false DeclareOperation( "IsEqualForCacheForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); ################################### ## #! @Section Transport Operations ## ################################### ## mor: x -> y ## equality_source: x -> x' ## equality_range: y -> y' ## TransportHom( mor, equality_source, equality_range ): x' -> y' DeclareOperation( "TransportHom", [ IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryMorphism ] ); ################################### ## #! @Section IsHomSetInhabited ## ################################### #! @Description #! The arguments are two objects <A>A</A> and <A>B</A>. #! The output is <C>true</C> if there exists a morphism from <A>A</A> to <A>B</A>, #! otherwise the output is <C>false</C>. #! @Arguments A, B #! @Returns a boolean DeclareOperation( "IsHomSetInhabited", [ IsCapCategoryObject, IsCapCategoryObject ] ); ################################### ## #! @Section SetOfMorphisms ## ################################### #! @Description #! Return a duplicate free list of morphisms of the finite category <A>C</A>. #! @Arguments C #! @Returns a list of a &CAP; category morphisms DeclareAttribute( "SetOfMorphismsOfFiniteCategory", IsCapCategory ); #! @Description #! Return a duplicate free list of morphisms of the finite category <A>C</A>. #! The corresponding &CAP; operation is <C>SetOfMorphismsOfFiniteCategory</C>. #! @Arguments C #! @Returns a list of &CAP; category objects DeclareAttribute( "SetOfMorphisms", IsCapCategory ); ################################### ## #! @Section Homomorphism structures ## ################################### #! Homomorphism structures are way to "oversee" the homomorphisms between two given objects. #! Let $C$, $D$ be categories. #! A $D$-homomorphism structure for $C$ consists of the following data: #! * a functor $H: C^{\mathrm{op}} \times C \rightarrow D$ (when $C$ and $D$ are Ab-categories, #! * an object $1 \in D$, called the distinguished object, #! * a bijection $\nu: \mathrm{Hom}_{C}(a,b) \simeq \mathrm{Hom}_{D}(1, H(a,b))$ natural i #! @Description #! The arguments are two objects $a, b$ in $C$. #! The output is the value of the homomorphism structure on objects $H(a,b)$. #! @Returns an object in $D$ #! @Arguments a,b DeclareOperation( "HomomorphismStructureOnObjects", [ IsCapCategoryObject, IsCapCategoryObject ] ); #! @Description #! The arguments are two morphisms $\alpha: a \rightarrow a', \beta: b \rightarrow b'$ in $C$. #! The output is the value of the homomorphism structure on morphisms $H(\alpha, \beta )$. #! @Returns a morphism in $\mathrm{Hom}_{D}(H(a',b), H(a,b'))$ #! @Arguments alpha, beta DeclareOperation( "HomomorphismStructureOnMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! The arguments are an object $s = H(a',b)$ in $D$, #! two morphisms $\alpha: a \rightarrow a', \beta: b \rightarrow b'$ in $C$, #! and an object $r = H(a,b')$ in $D$. #! The output is the value of the homomorphism structure on morphisms $H(\alpha, \beta )$. #! @Returns a morphism in $\mathrm{Hom}_{D}(H(a',b), H(a,b'))$ #! @Arguments s, alpha, beta, r DeclareOperation( "HomomorphismStructureOnMorphismsWithGivenObjects", [ IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryOb #! @Description #! The argument is a category $C$. #! The output is the distinguished object $1$ in $D$ of the homomorphism structure. #! @Returns an object in $D$ #! @Arguments C DeclareAttribute( "DistinguishedObjectOfHomomorphismStructure", IsCapCategory ); #! @Description #! The argument is a morphism $\alpha: a \rightarrow a'$ in $C$. #! The output is the corresponding morphism #! $\nu( \alpha ): 1 \rightarrow H(a,a')$ in $D$ of the homomorphism structure. #! @Returns a morphism in $\mathrm{Hom}_{D}(1, H(a,a'))$ #! @Arguments alpha DeclareAttribute( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphi IsCapCategoryMorphism ); #! @Description #! The arguments are the distinguished object $1$, a morphism $\alpha: a \rightarrow a'$, and t #! The output is the corresponding morphism #! $\nu( \alpha ): 1 \rightarrow r$ in $D$ of the homomorphism structure. #! @Returns a morphism in $\mathrm{Hom}_{D}(1, r)$ #! @Arguments distinguished_object, alpha, r DeclareOperation( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphi [ IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryObject ] ); #! @Description #! The arguments are #! objects $a,a'$ in $C$ #! and a morphism $\iota: 1 \rightarrow H(a,a')$ in $D$. #! The output is the corresponding morphism #! $\nu^{-1}(\iota): a \rightarrow a'$ in $C$ of the homomorphism structure. #! @Returns a morphism in $\mathrm{Hom}_{C}(a,a')$ #! @Arguments a,a',iota DeclareOperation( "InterpretMorphismFromDistinguishedObjectToHomomorphismStructur [ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism ] ); #! @Description #! The arguments are three lists $\alpha$, $\beta$, and $\gamma$. #! The first list $\alpha$ (the left coefficients) is a list of list of morphisms $\alpha_{ij}: #! where $i = 1 \dots m$ and $j = 1 \dots n$ for integers $m,n \geq 1$. #! The second list $\beta$ (the right coefficients) is a list of list of morphisms $\beta_{ij}: #! where $i = 1 \dots m$ and $j = 1 \dots n$. #! The third list $\gamma$ (the right side) is a list of morphisms $\gamma_i: A_i \rightarrow D_ #! where $i = 1, \dots, m$. #! Assumes that a solution to the linear system defined by $\alpha$, $\beta$, $\gamma$ exists, #! there exist morphisms $X_j: B_j \rightarrow C_j$ for $j=1\dots n$ such that #! $\sum_{j = 1}^n \alpha_{ij}\cdot X_j \cdot \beta_{ij} = \gamma_i$ #! for all $i = 1 \dots m$. #! The output is list of such morphisms $X_j: B_j \rightarrow C_j$ for $j=1\dots n$. #! @Returns a list of morphisms $[X_1, \dots, X_n]$ #! @Arguments alpha, beta, gamma DeclareOperation( "SolveLinearSystemInAbCategory", [ IsList, IsList, IsList ] ); #! @Description #! This is a convenience operation. #! Like <C>SolveLinearSystemInAbCategory</C>, #! but without the assumption that a solution exists. #! If no solution exists, `fail` is returned. #! @Returns a list of morphisms $[X_1, \dots, X_n]$ or `fail` #! @Arguments alpha, beta, gamma DeclareOperation( "SolveLinearSystemInAbCategoryOrFail", [ IsList, IsList, IsList ] ); #! @Description #! Like <C>SolveLinearSystemInAbCategory</C>, #! but the output is simply <C>true</C> if a solution exists, #! <C>false</C> otherwise. #! @Returns a boolean #! @Arguments alpha, beta, gamma DeclareOperation( "MereExistenceOfSolutionOfLinearSystemInAbCategory", [ IsList, IsList, IsList ] ); #! @Description #! Like <C>SolveLinearSystemInAbCategory</C>, #! but the output is simply <C>true</C> if a unique solution exists, #! and <C>false</C> otherwise. #! @Returns a boolean #! @Arguments alpha, beta, gamma DeclareOperation( "MereExistenceOfUniqueSolutionOfLinearSystemInAbCategory", [ IsList, IsList, IsList ] ); #! @Description #! Like <C>SolveLinearSystemInAbCategory</C> #! but the output is <C>true</C> if the homogeneous system has only the trivial zero solution, #! and <C>false</C> otherwise. #! @Returns a boolean #! @Arguments alpha, beta DeclareOperation( "MereExistenceOfUniqueSolutionOfHomogeneousLinearSystemInAbCate [ IsList, IsList ] ); #! @Description #! The arguments are two lists of lists $\alpha$ and $\beta$ of morphisms in a linear category ov #! The first list $\alpha$ (the left coefficients) is a list of list of morphisms $\alpha_{ij}: #! where $i = 1, \dots, m$ and $j = 1, \dots, n$ for integers $m,n \geq 1$. #! The second list $\beta$ (the right coefficients) is a list of list of morphisms $\beta_{ij}: #! where $i = 1, \dots, m$ and $j = 1, \dots, n$. #! The output is a generating set $[X^1,\dots,X^t]$ for the solutions of the homogeneous linea #! $\sum_{j = 1}^n \alpha_{ij}\cdot X_{j} \cdot \beta_{ij} = 0, ~i = 1, \dots, m$. #! Particularly, each $X^k$ is a list (of length $n$) of morphisms $X^k_j:B_j \to C_j, j=1,\dot #! @Returns a list of lists of morphisms $[X^1, \dots, X^t]$ #! @Arguments alpha, beta DeclareOperation( "BasisOfSolutionsOfHomogeneousLinearSystemInLinearCategory", [ IsList, IsList ] ); #! @Description #! The arguments are four lists of lists $\alpha$, $\beta$, $\gamma$, $\delta$ of morphisms in #! Each of $\alpha$ and $\gamma$ is a list of list of morphisms $\alpha_{ij}, \gamma_{ij}: A_i \ #! where $i = 1, \dots, m$ and $j = 1, \dots, n$ for integers $m,n \geq 1$. #! Each of $\beta$ and $\delta$ is also a list of list of morphisms $\beta_{ij}, \delta_{ij}: C_ #! where $i = 1, \dots, m$ and $j = 1, \dots, n$. #! The output is a generating set $[X^1,\dots,X^t]$ for the solutions of the homogeneous linea #! defined by $\alpha$, $\beta$, $\gamma$ and $\delta$, i.e., #! $\sum_{j = 1}^n \alpha_{ij}\cdot X^{k}_{j} \cdot \beta_{ij} = \sum_{j = 1}^n \gamma_{ij}\ #! for all $i = 1, \dots, m$ and all $k = 1, \dots, t$. #! @Returns a list of lists of morphisms $[X^1, \dots, X^t]$ #! @Arguments alpha, beta, gamma, delta DeclareOperation( "BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategor [ IsList, IsList, IsList, IsList ] ); #! @Description #! The arguments are two lists of lists $\alpha$, $\delta$ morphisms in some linear category #! over commutative ring. #! $\alpha$ is a list of list of morphisms $\alpha_{ij}:A_i \rightarrow B_j$ and #! $\delta$ is a list of list of morphisms $\delta_{ij}:C_j \rightarrow D_i$, #! where $i = 1, \dots, m$ and $j = 1, \dots, n$ for integers $m,n \geq 1$. #! The method delegates to <C>BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCa #! * $\beta_{ij}$ equals <C>IdentityMorphism</C>(<C>Source</C>($\delta_{ij}$)) if $\delta_{ij} #! and <C>ZeroMorphism</C>(<C>Source</C>($\delta_{ij}$), <C>Range</C>($\delta_{ij}$)) otherwise, #! * $\gamma_{ij}$ equals <C>IdentityMorphism</C>(<C>Source</C>($\alpha_{ij}$)) if $\alpha_{ij #! and <C>ZeroMorphism</C>(<C>Source</C>($\alpha_{ij}$), <C>Range</C>($\alpha_{ij}$)) otherwise; #! for all $i = 1, \dots, m$ and $j = 1, \dots, n$. #! @Returns a list of lists of morphisms $[X^1, \dots, X^t]$ #! @Arguments alpha, delta DeclareOperation( "BasisOfSolutionsOfHomogeneousDoubleLinearSystemInLinearCategor [ IsList, IsList ] ); #! @Description #! This is a convenience method. #! The arguments are two morphisms $\alpha: a \rightarrow a', \beta: b \rightarrow b'$ in $C$. #! The output is <C>HomomorphismStructureOnMorphisms</C> called on $\alpha$, $\beta$. #! @Returns a morphism in $\mathrm{Hom}_{D}(H(a',b), H(a,b'))$ #! @Arguments alpha, beta DeclareOperation( "HomStructure", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience method. #! The arguments are a morphism $\alpha: a \rightarrow a'$ and an object $b$ in $C$. #! The output is <C>HomomorphismStructureOnMorphisms</C> called on $\alpha$, $\mathrm{id}_b$ #! @Returns a morphism in $\mathrm{Hom}_{D}(H(a',b), H(a,b))$ #! @Arguments alpha, b DeclareOperation( "HomStructure", [ IsCapCategoryMorphism, IsCapCategoryObject ] ); #! @Description #! This is a convenience method. #! The arguments are an object $a$ and a morphism $\beta: b \rightarrow b'$ in $C$. #! The output is <C>HomomorphismStructureOnMorphisms</C> called on $\mathrm{id}_a$, $\beta$. #! @Returns a morphism in $\mathrm{Hom}_{D}(H(a,b), H(a,b'))$ #! @Arguments a, beta DeclareOperation( "HomStructure", [ IsCapCategoryObject, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience method. #! The arguments are two objects $a$ and $b$ in $C$. #! The output is <C>HomomorphismStructureOnObjects</C> called on $a,b$. #! @Returns an object #! @Arguments a, b DeclareOperation( "HomStructure", [ IsCapCategoryObject, IsCapCategoryObject ] ); #! @Description #! This is a convenience method for #! <C>InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure</C>. DeclareOperation( "HomStructure", [ IsCapCategoryMorphism ] ); #! @Description #! This is a convenience method for #! <C>InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism</C>. DeclareOperation( "HomStructure", [ IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism ] ); #! @Description #! This is a convenience method for #! <C>DistinguishedObjectOfHomomorphismStructure</C>. DeclareOperation( "HomStructure", [ IsCapCategory ] ); #! @BeginGroup #! @Description #! If $\iota\colon D \to E$ is a full embedding of categories, every $D$-homomorphism structur #! extends to a $E$-homomorphism structure for $C$. This operations accepts four functions #! and installs operations `DistinguishedObjectOfHomomorphismStructureExtendedByFull #! `HomomorphismStructureOnObjectsExtendedByFullEmbedding` etc. which correspond to th #! Note: To distinguish embeddings in different categories, in addition to $C$ also $E$ is pass #! When using this with different embeddings with the range category $E$, only the last embeddi #! The arguments are: #! * `object_function` gets the categories $C$ and $E$ and an object in $D$. #! * `morphism_function` gets the categories $C$ and $E$, an object in $E$, a morphism in $D$ and #! The objects are the results of `object_function` applied to the source and range of the morph #! * `object_function_inverse` gets the categories $C$ and $E$ and an object in $E$. #! * `morphism_function_inverse` gets the categories $C$ and $E$, an object in $D$, a morphism #! The objects are the results of `object_function_inverse` applied to the source and range of #! `object_function` and `morphism_function` define the embedding. `object_function_inv #! the inverse of the embedding on its image. #! @Returns nothing #! @Arguments C, E, object_function, morphism_function, object_function_inverse, morphi DeclareOperation( "ExtendRangeOfHomomorphismStructureByFullEmbedding", [ IsCapCategory, IsCapCategory, IsFunction, IsFunction, IsFunction, IsFunction ] ); #! @Arguments C, E, a, b DeclareOperation( "HomomorphismStructureOnObjectsExtendedByFullEmbedding", [ IsCapCategory, IsCapCategory, IsCapCategoryObject, IsCapCategoryObject ] ); #! @Arguments C, E, alpha, beta DeclareOperation( "HomomorphismStructureOnMorphismsExtendedByFullEmbedding", [ IsCapCategory, IsCapCategory, IsCapCategoryMorphism, IsCapCategoryMorphism ] ); #! @Arguments C, E, s, alpha, beta, r DeclareOperation( "HomomorphismStructureOnMorphismsWithGivenObjectsExtendedByFull [ IsCapCategory, IsCapCategory, IsCapCategoryObject, IsCapCategoryMorphism, IsCapCate #! @Arguments C, E DeclareOperation( "DistinguishedObjectOfHomomorphismStructureExtendedByFullEmbedd [ IsCapCategory, IsCapCategory ] ); #! @Arguments C, E, alpha DeclareOperation( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphi [ IsCapCategory, IsCapCategory, IsCapCategoryMorphism ] ); #! @Arguments C, E, distinguished_object, alpha, r DeclareOperation( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphi [ IsCapCategory, IsCapCategory, IsCapCategoryObject, IsCapCategoryMorphism, IsCapCate #! @Arguments C, E, a, a', iota DeclareOperation( "InterpretMorphismFromDistinguishedObjectToHomomorphismStructur [ IsCapCategory, IsCapCategory, IsCapCategoryObject, IsCapCategoryObject, IsCapCatego #! @EndGroup CapJitAddTypeSignature( "HomomorphismStructureOnObjectsExtendedByFullEmbedding", [ return CapJitDataTypeOfObjectOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "HomomorphismStructureOnMorphismsExtendedByFullEmbedding" return CapJitDataTypeOfMorphismOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "HomomorphismStructureOnMorphismsWithGivenObjectsExtended return CapJitDataTypeOfMorphismOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "DistinguishedObjectOfHomomorphismStructureExtendedByFull return CapJitDataTypeOfObjectOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomo return CapJitDataTypeOfMorphismOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "InterpretMorphismAsMorphismFromDistinguishedObjectToHomo return CapJitDataTypeOfMorphismOfCategory( input_types[2].category ); end ); CapJitAddTypeSignature( "InterpretMorphismFromDistinguishedObjectToHomomorphismSt return CapJitDataTypeOfMorphismOfCategory( input_types[1].category ); end ); #! @Description #! Chooses the identity on $D$ as the full embedding in #! <Ref Oper="ExtendRangeOfHomomorphismStructureByFullEmbedding" Label="for IsCapCate #! This is useful to handle this case as a degenerate case of #! <Ref Oper="ExtendRangeOfHomomorphismStructureByFullEmbedding" Label="for IsCapCate #! @Returns nothing #! @Arguments C DeclareOperation( "ExtendRangeOfHomomorphismStructureByIdentityAsFullEmbedding", [ IsCapCategory ] ); #! @Description #! The argument are two objects <A>a</A>, <A>b</A>. #! The output is a list of all morphisms from <A>a</A> to <A>b</A>. #! @Returns a list of morphisms in $\mathrm{Hom}( a, b )$ #! @Arguments a, b DeclareOperation( "MorphismsOfExternalHom", [ IsCapCategoryObject, IsCapCategoryObject ] ); #! @Description #! The arguments are objects $a,b$ in a $k$-linear category $C$. #! The output is a list $L$ of morphisms which is a basis of $\mathrm{Hom}_{C}(a,b)$ in #! the sense that any given morphism $\alpha: a \to b$ can uniquely be written as a #! linear combination of $L$ with the coefficients in #! <C>CoefficientsOfMorphism</C>($\alpha$). #! @Returns a list of morphisms in $\mathrm{Hom}_{C}(a,b)$ #! @Arguments a, b DeclareOperation( "BasisOfExternalHom", [ IsCapCategoryObject, IsCapCategoryObject ] ); #! @Description #! This is a convenience method. #! The argument is a morphism $\alpha: a \to b$ in a $k$-linear category $C$. #! The output is a list of coefficients of $\alpha$ with respect to the list #! <C>BasisOfExternalHom</C>(<A>a</A>,<A>b</A>). #! @Returns a list of elements in $k$ #! @Arguments alpha DeclareAttribute( "CoefficientsOfMorphism", IsCapCategoryMorphism ); ################################### ## #! @Section Simplified Morphisms ## ################################### #! Let $\phi: A \rightarrow B$ be a morphism. #! There are several different natural ways to look at $\phi$ as an object in an ambient category #! - $\mathrm{Hom}( A, B )$, the set of homomorphisms with the equivalence relation $\mathtt{I #! - $\sum_{A}\mathrm{Hom}( A, B )$, the category of morphisms where the range is fixed, #! - $\sum_{B}\mathrm{Hom}( A, B )$, the category of morphisms where the source is fixed, #! - $\sum_{A,B}\mathrm{Hom}( A, B )$, the category of morphisms where neither source nor rang #! and furthermore, if $\phi$ happens to be an endomorphism $A \rightarrow A$, #! we also have #! - $\sum_{A}\mathrm{Hom}(A,A)$, the category of endomorphisms. #! Let $\mathbf{C}$ be one of the categories above in which $\phi$ may reside as an object, #! and let $i$ be a non-negative integer or $\infty$. #! CAP provides commands for passing from $\phi$ to $\phi_i$, where $\phi_i$ is isomorphic to $ #! in $\mathbf{C}$, but "simpler". #! The idea is that the greater the $i$, the "simpler" the $\phi_i$ (but this could mean the harde #! with $\infty$ as a possible value. #! The case $i = 0$ defaults to the identity operator for all simplifications. #! For the Add-operatations, only the cases $i \geq 1$ have to be given as functions. #! #! #! $\ $ #! #! #! If we regard $\phi$ as an object in the category $\mathrm{Hom}( A, B )$, #! $\phi_i$ is again in $\mathrm{Hom}( A, B )$ such that $\phi \sim_{A,B} \phi_i$. #! This case is handled by the following commands: #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is a simplified morphism $\phi_i$. #! @Returns a morphism in $\mathrm{Hom}(A,B)$ #! @Arguments phi, i DeclareOperation( "SimplifyMorphism", [ IsCapCategoryMorphism, IsObject ] ); #! $\ $ #! #! If we regard $\phi$ as an object in the category $\sum_{A}\mathrm{Hom}( A, B )$, #! then $\phi_i$ is a morphism of type $A_i \rightarrow B$ and there is an isomorphism #! $\sigma_i: A \rightarrow A_i$ such that #! $\phi_i \circ \sigma_i \sim_{A,B} \phi$. #! This case is handled by the following commands: ## SimplifySource #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is a simplified morphism with simplified source $\phi_i: A_i \rightarrow B$. #! @Returns a morphism in $\mathrm{Hom}(A_i,B)$ #! @Arguments phi, i DeclareOperation( "SimplifySource", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $(\sigma_i)^{-1}: A_i \rightarrow A$. #! @Returns a morphism in $\mathrm{Hom}(A_i,A)$ #! @Arguments phi, i DeclareOperation( "SimplifySource_IsoToInputObject", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $\sigma_i: A \rightarrow A_i$. #! @Returns a morphism in $\mathrm{Hom}(A,A_i)$ #! @Arguments phi, i DeclareOperation( "SimplifySource_IsoFromInputObject", [ IsCapCategoryMorphism, IsObject ] ); ## SimplifyRange #! $\ $ #! #! If we regard $\phi$ as an object in the category $\sum_{B}\mathrm{Hom}( A, B )$, #! then $\phi_i$ is a morphism of type $A \rightarrow B_i$ and there is an isomorphism #! $\rho_i: B \rightarrow B_i$ such that #! $ \rho_i^{-1} \circ \phi_i\sim_{A,B} \phi$. #! This case is handled by the following commands: #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is a simplified morphism with simplified range $\phi_i: A \rightarrow B_i$. #! @Returns a morphism in $\mathrm{Hom}(A,B_i)$ #! @Arguments phi, i DeclareOperation( "SimplifyRange", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $(\rho_i)^{-1}: B_i \rightarrow B$. #! @Returns a morphism in $\mathrm{Hom}(B_i,B)$ #! @Arguments phi, i DeclareOperation( "SimplifyRange_IsoToInputObject", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $\rho_i: B \rightarrow B_i$. #! @Returns a morphism in $\mathrm{Hom}(B,B_i)$ #! @Arguments phi, i DeclareOperation( "SimplifyRange_IsoFromInputObject", [ IsCapCategoryMorphism, IsObject ] ); ## SimplifySourceAndRange* #! $\ $ #! #! If we regard $\phi$ as an object in the category $\sum_{A, B}\mathrm{Hom}( A, B )$, #! then $\phi_i$ is a morphism of type $A_i \rightarrow B_i$ and there is are isomorphisms #! $\sigma_i: A \rightarrow A_i$ and #! $\rho_i: B \rightarrow B_i$ such that #! $ \rho_i^{-1} \circ \phi_i \circ \sigma_i \sim_{A,B} \phi$. #! This case is handled by the following commands: #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is a simplified morphism with simplified source and range $\phi_i: A_i \rightarr #! @Returns a morphism in $\mathrm{Hom}(A_i,B_i)$ #! @Arguments phi, i DeclareOperation( "SimplifySourceAndRange", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $(\rho_i)^{-1}: B_i \rightarrow B$. #! @Returns a morphism in $\mathrm{Hom}(B_i,B)$ #! @Arguments phi, i DeclareOperation( "SimplifySourceAndRange_IsoToInputRange", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $\rho_i: B \rightarrow B_i$. #! @Returns a morphism in $\mathrm{Hom}(B,B_i)$ #! @Arguments phi, i DeclareOperation( "SimplifySourceAndRange_IsoFromInputRange", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $(\sigma_i)^{-1}: A_i \rightarrow A$. #! @Returns a morphism in $\mathrm{Hom}(A_i,A)$ #! @Arguments phi, i DeclareOperation( "SimplifySourceAndRange_IsoToInputSource", [ IsCapCategoryMorphism, IsObject ] ); #! @Description #! The arguments are a morphism $\phi: A \rightarrow B$ and a non-negative integer $i$ or <C>infin #! The output is the isomorphism $\sigma_i: A \rightarrow A_i$. #! @Returns a morphism in $\mathrm{Hom}(A,A_i)$ #! @Arguments phi, i DeclareOperation( "SimplifySourceAndRange_IsoFromInputSource", [ IsCapCategoryMorphism, IsObject ] ); ## SimplifyEndo* #! $\ $ #! #! If $\phi:A \rightarrow A$ is an endomorphism, we may regard it as an object in the category $\s #! In this case #! $\phi_i$ is a morphism of type $A_i \rightarrow A_i$ and there is an isomorphism #! $\sigma_i: A \rightarrow A_i$ such that --> -------------------- --> maximum size reached --> -------------------- [ Verzeichnis aufwärts0.29unsichere Verbindung Übersetzung europäischer Sprachen durch Browser ] |
2026-03-28