‣ IsQuotientCategory( C ) | ( filter ) |
Returns: a boolian
The Gap filter of the Cap quotient categories.
‣ IsQuotientCategoryMorphism( alpha ) | ( filter ) |
Returns: a boolian
The gap category of quotient categories morphisms
‣ IsQuotientCategoryCell( a, object ) | ( filter ) |
Returns: true or false
The GAP category of cells in the quotient category.
‣ IsQuotientCategoryObject( a, object ) | ( filter ) |
Returns: true or false
The GAP category of objects in the quotient category.
‣ QuotientCategory( C, F ) | ( operation ) |
Returns: a Cap category
The input is a category \(C\) and a function \(F\). For two objects \(a\) and \(b\) in \(C\), the function \(F\) can be applied on two morphisms \(\alpha,\beta \in \mathrm{Hom}_C(a,b)\) and returns true if \(\alpha \sim \beta\) and false otherwise. The output is the quotient category \(C/F\).
‣ QuotientCategoryMorphism( _a_, alpha, _b_ ) | ( operation ) |
Returns: a morphism
The arguments are two objects \(\underline{a}, \underline{b}\) in a quotient category \(Q=C/F\) and a morphism \(\alpha:a \rightarrow b\) in \(C\). The output is \(\underline{\alpha}: \underline{a} \rightarrow \underline{b}\) in \(Q\).
‣ QuotientCategoryMorphism( Q, alpha ) | ( operation ) |
Returns: a morphism
The input is a quotient category \(Q=C/F\) and a morphism \(\alpha:a \rightarrow b\) in \(C\). The output is \(\underline{\alpha}: \underline{a} \rightarrow \underline{b}\) in \(Q\).
1.2-4 \/‣ \/( Q, alpha ) | ( operation ) |
Returns: a morphism
The input is a quotient category \(Q=C/F\) and a morphism \(\alpha:a \rightarrow b\) in \(C\). The output is \(\underline{\alpha}: \underline{a} \rightarrow \underline{b}\) in \(Q\).
‣ QuotientCategoryObject( Q, a ) | ( operation ) |
Returns: an object
The input is a quotient category \(Q=C/F\) and an object \(a \in C\). The output is \(\underline{a} \in Q\).
1.2-6 \/‣ \/( a, Q ) | ( operation ) |
Returns: an object
The input is a quotient category \(Q=C/F\) and an object \(a \in C\). The output is \(\underline{a} \in Q\).
‣ UnderlyingCategory( Q ) | ( attribute ) |
Returns: a category
The input is a quotient category \(Q := C/F\). The output is the category \(C\).
‣ CongruencyTestFunctionForQuotientCategory( Q ) | ( attribute ) |
Returns: a function
The input is a quotient category \(Q := C/F\). The output is the congruence test function \(F\).
‣ ProjectionFunctor( Q ) | ( attribute ) |
Returns: a functor
The input is a quotient category \(Q := C/F\). The output is the canonical projection functor \(\pi:C \rightarrow C/F\).
‣ UnderlyingCell( _alpha_ ) | ( attribute ) |
Returns: a morphism
The input is a morphism \(\underline{\alpha}: \underline{a} \rightarrow \underline{b}\) in some quotient category \(Q=C/F\) and the output is \(\alpha: a \rightarrow b\) in \(C\).
‣ UnderlyingCell( _a_ ) | ( attribute ) |
Returns: an object
The input is an object \(\underline{a}\) in some quotient category \(Q=C/F\) and the output is \(a\) in \(C\).
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